Weak-strong uniqueness for energy-reaction-diffusion systems
aa r X i v : . [ m a t h . A P ] F e b WEAK-STRONG UNIQUENESS FORENERGY-REACTION-DIFFUSION SYSTEMS
KATHARINA HOPF
Abstract.
We establish weak-strong uniqueness and stability properties of renormal-ised solutions to a class of energy-reaction-diffusion systems, which genuinely featurecross-diffusion effects. The systems considered are motivated by thermodynamicallyconsistent models, and their formal entropy structure allows us to use as a key toola suitably adjusted relative entropy method. Weak-strong uniqueness is obtained forgeneral entropy-dissipating reactions without growth restrictions, and certain modelswith a non-integrable diffusive flux. The results also apply to a class of (isoenergetic)reaction-cross-diffusion systems. Introduction
It is well acknowledged that the evolution of a system of diffusing and reacting chem-icals is influenced by the thermal state of the system.
Energy-reaction-diffusion systems (ERDS) take into account this thermal dependency by consistently coupling the evolutionof the chemical concentrations c = ( c , . . . , c n ) to a heat-type equation for the internal en-ergy density u . Choosing the internal energy density as the thermal variable (as opposedto temperature for instance) has the advantage that the underlying physical entropy isjointly concave in the state variables z = ( u, c ) [34, 35].Recently, global existence of weak and renormalised solutions has been obtained for aclass of thermodynamically consistent ERDS [24] taking the form (with z := ( u, c )) ∂ t u = ∇ · (cid:0) A j ( z ) ∇ z j (cid:1) , t > , x ∈ Ω ,∂ t c i = ∇ · (cid:0) A ij ( z ) ∇ z j (cid:1) + R i ( z ) , t > , x ∈ Ω , i ∈ { , . . . , n } , A ij ( z ) ∇ z j · ν, t > , x ∈ ∂ Ω , i ∈ { , . . . , n } , (1.1)see also the more explicit system (2.5). Eq. (1.1) is supplemented with an initial con-dition ( u, c ) | t =0 = ( u in , c in ) for x ∈ Ω, where Ω ⊂ R d is a bounded Lipschitz domainwith outer unit normal ν . Note that we use the summation convention omitting thesummation symbol in repeatedly occurring indices (here P nj =0 ). The diffusion matrix A ( z ) = ( A ij ( z )) i,j =0 ,...,n and the reactions ( R i ( z )) i =1 ,...,n are obtained from an underlyingformal gradient structure based on entropy functionals H ( z ) = ´ Ω h ( z ) d x with convexdensities h ( z ) = h ( u, c ) taking the form h ( u, c , . . . , c n ) = − σ ( u ) + n X i =1 b ( c i , w i ( u )) . Here, σ denotes the thermal part when the concentrations c i are in their thermodynamicequilibrium w i = w i ( u ), and b ( s, e ) := eλ ( s/e ) with λ denoting the Boltzmann function,see Section 1.1 for details. The absence of reactions in the u -component of (1.1) reflectsthe property of conservation of the total (internal) energy ´ Ω u . Mathematics Subject Classification.
Key words and phrases.
Energy-reaction-diffusion systems, weak-strong uniqueness, entropy method,convexity method, renormalized solutions, cross diffusion.
ERDS genuinely feature cross-diffusion effects, such as concentration flux driven bygradients of the internal energy density and energy flux due to concentration gradients,which are one of the main sources of difficulties in their analysis. These phenomena areclosely linked to the thermodynamic origin of ERDS, and are related to the
Soret effect and the
Dufour effect , well-known in physics, which describe concentration flux due totemperature gradients resp. heat flux driven by concentration gradients.In the present manuscript, we aim to derive stability properties including a weak-stronguniqueness result for ERDS (1.1) based on their thermodynamic structure. This can beseen as a way of justifying the weak solution concept in [24]. The main contributionof [24] was to identify non-trivial classes of thermodynamically consistent models that al-low for an existence theory of generalised solutions. Interestingly, even in a cross-diffusiondominant regime and without physically restrictive growth conditions on the reactions,existence has been obtained in [24] based on the notion of renormalised solutions [22]. Ourweak-strong uniqueness principle covers models involving various cross-diffusion phenom-ena (Soret effect, Dufour effect, cross diffusion between species), and applies in particularto a class of isoenergetic reaction-cross-diffusion systems, thus generalising [12]. Weak-strong uniqueness is obtained from a weak stability estimate for a generalised distanceinvolving as in [23] an adjusted relative entropy. By suitably exploiting the thermody-namic structure of the system some of the technical issues arising in the proof of [12] willbe avoided. We also obtain an asymptotic stability result.1.1.
Thermodynamic modelling.
Let us now briefly specify the thermodynamic struc-ture considered in the present manuscript. For more background on the modelling, werefer to [24, 39, 28, 36]. Models compatible with thermodynamics can be derived usingthe Onsager formalism in [36]. Here, the main ingredient is a triple (
Z, H, K ) consistingof a state space Z , a driving functional H , and a so-called Onsager operator K . Typic-ally, Z ⊂ X is a convex subset of a Banach space X , H : Z → R ∪ {∞} a differentiableand convex functional on Z (below usually referred to as entropy due to its correspond-ence to the negative of the physical entropy), while K can be seen as a generalised inverseRiemannian metric tensor on Z . More specifically, for every z ∈ Z , K ( z ) defines a positivesemi-definite and symmetric (unbounded) operator from T ∗ z Z to T z Z . If G = K − exists,the triple ( Z, H, G ) forms a gradient system. Then, motivated by the classical gradientflow equation G ( z ) ˙ z = − DH ( z ), with ˙ z denoting the time derivative of z = z ( t ), oneconsiders the evolution law ˙ z = − K ( z ) DH ( z ) , where here DH denotes the Fr´echet derivative of the functional H . An advantage of thisOnsager form is that it facilitates the consistent coupling of different physical phenomena,which can be realised by an additive decomposition of K [36]. Observe that, formally,the above structure encodes the following core entropy dissipation property dd t H ( z ) = −h DH ( z ) , K ( z ) DH ( z ) i ≤ z = z ( t ) of the above law. Conservation of the total energy E ( z ) , with E denoting the energy functional on Z , can be guaranteed by imposing thecondition K DE = K ∗ DE ! = 0, which implies that dd t E ( z ) = −h DE ( z ) , K ( z ) DH ( z ) i = 0 . In the context of ERDS, we consider, as introduced above, z = ( u, c ) with u the internalenergy density and c = ( c , . . . , c n ) the vector of concentrations. We focus on entropiesof the form H ( z ) = ˆ Ω h ( z ) d x EAK-STRONG UNIQUENESS FOR ERDS 3 with densities h ( z ) = h ( u, c ) = − σ ( u ) + n X i =1 b ( c i , w i ( u )) , composed of a thermal part σ ( u ) and a relative Boltzmann entropy b ( s, e ) = eλ ( s/e ),where λ ( r ) := r log( r ) − r + 1 , (1.3)and with w i = w i ( u ), i = 1 , . . . , n , denoting the thermodynamic equilibria of the concen-trations c i . The dependence of w i on the internal energy density u results in a strongcoupling of the system and is one of the main sources of difficulties in the analysis. Itwill be convenient to introduce the function ˆ σ ( u ) = σ ( u ) − P ni =1 w i ( u ) + n and rewrite h ( u, c ) in the following more explicit form h ( u, c ) = − ˆ σ ( u ) + n X i =1 (cid:16) λ ( c i ) − c i log w i ( u ) (cid:17) . (h1)To simplify the exposition, we will impose the following concrete conditions on the coef-ficient functions (cf. [24]):ˆ σ ∈ C ((0 , ∞ )) strictly concave & non-decreasing. w i ∈ C ([0 , ∞ )) ∩ C ((0 , ∞ )) concave & non-decr. with w i (0) > i .lim u ↓ ˆ σ ′ ( u ) = + ∞ , lim u ↑∞ ˆ σ ′ ( u ) = 0; sup u ∈ (0 , ˆ σ ′′ ( u ) < ∃ β ∈ (0 ,
1) such that w i ( u ) . u β for all i ∈ { , . . . , n } . (h2)Typical choices are ˆ σ ( u ) = a log( u ) or ˆ σ ( u ) = au ν for some ν ∈ (0 , a >
0, and w i ( u ) = ( b i, u + b i, ) β i or w i ( u ) = b i, u β i + b i, for β i ∈ (0 , , b i, > , b i, ≥ In fact, in the proof of our main theorem (Thm 2.8),identity (h1) is only used to guarantee the coercivity properties in Proposition 3.2. Thecrucial point in the assumptions (h2) on the coefficient functions is that they ensure goodconvexity properties, and more specifically the locally uniform positive definiteness ofthe Hessian D h , which is essential for estimate (3.5) in Prop. 3.2. The monotonicityassumptions on ˆ σ and w i are relevant from the modelling point of view, since they ensurethat u h ( u, c ) is non-increasing, so that temperature, which is given by − ∂ u h , isnon-negative.As in [39, 24] we are primarily interested in Onsager operators K of the form K ( z ) ζ = K diff ( z ) ζ + K react ( z ) ζ = −∇ · ( M ( z ) ∇ ζ ) + L ( z ) ζ , where M ( z ) , L ( z ) ∈ R (1+ n ) × (1+ n ) are positive semi-definite symmetric matrices and where ∇ = ∇ x is the gradient with respect to x ∈ Ω. We will complement K with the no-flux boundary conditions M ∇ ζ · ν = 0 on ∂ Ω, where ν denotes the outer unit normalvector to ∂ Ω. Observing that E ( u, c ) = ´ Ω u describes the total (internal) energy, thecondition K DE ≡
0, ensuring energy conservation, means that ker L ( z ) ⊇ span { (1 , T } .Thus, by the symmetry of L , the zeroth component R of R ( z ) := − L ( z ) Dh ( z ) vanishes.Moreover, positive semi-definiteness of L implies the inequality D i h ( z ) R i ( z ) ≤ . (1.4)In this paper, the specific form of L ( z ) will not be relevant. Instead, we directly workwith reactions R ( z ) of the form R ( z ) := (0 , R ( z ) , . . . , R n ( z )) See Sec. 2.3.2 for an example of a different entropy density that our technique can be adapted to.
K. HOPF satisfying (1.4).With K as above, the equation ˙ z = − K ( z ) DH ( z ) can be written in the form (1.1) bychoosing A ( z ) := M ( z ) D h ( z ) . In short, ∂ t z = ∇ · (cid:0) A ( z ) ∇ z (cid:1) + R ( z ) , t > , x ∈ Ω , A ( z ) ∇ z · ν, t > , x ∈ ∂ Ω , ( erds )subject to an initial condition z | t =0 = z in . In the above setting, the entropy dissipationproperty (1.2) takes the formdd t H ( z ) + ˆ Ω P ( z ) d x = ˆ Ω D i h ( z ) R i ( z ) d x ≤ , where P ( z ) := ∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) ≥
0, by the positive semi-definiteness of themobility matrix M , which will be assumed throughout. Supposing, for instance, that P ( z ) & P ni =1 |∇√ c i | (as it can be proved for many of the models considered in [24],see Section 2.3 and Lemma 6.1), and using conservation of ´ Ω u together with suitablebounds on H ( z ) (cf. Lemma 6.3), the entropy dissipation property provides a prioricontrol of P ni =1 k∇√ c i k L t,x . Let us further note that the fact that R ( z ) vanishes givesto some extent a scalar-like structure to the u -component of ( erds ), and if, for instance, M is chosen such that A j = a δ j for some function a = a ( z ) ≥
0, any L p -type energy p k u ( t ) k pL p , p ∈ (1 , ∞ ), is formally non-increasing in time.1.2. Motivation and strategy.
Being able, for a given PDE, to identify concepts ofsolutions for which both existence and uniqueness can be established is a fundamentalconcern in modelling and analysis. For scalar equations, there are various tools to identifyframeworks allowing for the existence of a unique solution, even in regimes of low regu-larity and with strong nonlinearities. One approach is based on the ‘doubling variables’technique first employed by Kruˇzkov [32] to entropy solutions of first-order equations,and extended by Carrillo [6] to hyperbolic-parabolic-elliptic equations. The concept wasadapted to situations where L ∞ bounds are not available to give uniqueness in a classof renormalised solutions [3, 7]. See also [2, 40, 1] for more recent developments. Let usalso mention the Young measure approach to conservation laws going back to Tartar [43]and DiPerna [19] who obtained, in the scalar case, uniqueness of solutions obeying an en-tropy inequality. For second order parabolic and elliptic equations the viscosity solutiontechnique and associated comparison principles [30, 14] are powerful tools and the key toa variety of wellposedness results in geometric, highly nonlinear or degenerate settings,see e.g. [20, 13, 5, 8]. Some extensions of the viscosity solution approach to systemsare available for weakly coupled problems with a monotonicity condition [29]. Furtheruniqueness results applying to specific systems and typically in more regular situationsinclude [25, 31, 10, 41, 26, 4].In general, the case of strongly coupled (parabolic) systems tends to be much moredifficult. While under a parabolicity condition the existence of suitable generalised solu-tions (here referred to as ‘weak’ solutions) can often be established, positive uniquenessresults in such general settings are rare. It is therefore common, to relax the quest foruniqueness to the problem of whether weak solutions are uniquely determined in situ-ations where a sufficiently regular solution (a ‘strong’ solution) happens to exist. In otherwords, one is interested in the question of whether such strong solutions are unique in apotentially much larger class of weak solutions. The question of weak-strong uniquenessis classical in fluid dynamics problems and goes back to Leray’s fundamental work [33],where it was established for the incompressible Navier–Stokes equations. We refer to the EAK-STRONG UNIQUENESS FOR ERDS 5 survey by Wiedemann [45] for more details and further references. For recent advanceson conditional uniqueness results for dissipative measure-valued solutions to conservationlaws, see [27] and references therein. Quite interesting in the thermodynamics context ismoreover the relative entropy technique employed by DiPerna [18] and Dafermos [16] forhyperbolic conservation laws.Relative entropy methods are nowadays a standard tool to study weak stability proper-ties of nonlinear systems endowed with a (convex) entropy structure. Generally speaking(using the notation introduced in Sec. 1.1), a relative entropy of the form H rel ( z, ˜ z ) = H ( z ) − ˆ Ω D i h (˜ z )( z i − ˜ z i ) d x − H (˜ z ) (1.5)is used to measure the distance between a weak solution z and a strong solution ˜ z . Observethat for convex entropies H ( z ), the map z H rel ( z, ˜ z ) is a non-negative, convex func-tional vanishing in z = ˜ z . The thermodynamic structure ensures that regular solutionsautomatically satisfy an entropy dissipation balance (cf. eq. (1.2)). Physically relevantprocesses may, however, in general possess less regularity and here an entropy inequalityis often added as an admissibility criterion for weak solutions. The goal is then to obtainan upper bound on the time evolution of the relative entropy that implies stability of aregular flow on a finite time horizon among generalised solutions. This means that givena regular flow, any weak solution that is initially close (in relative entropy) will remainclose for some time.In the present paper, we pursue such a strategy in the context of ERDS. Weak-stronguniqueness has recently been obtained for entropy-dissipating reaction-diffusion systemswith a uniformly elliptic and bounded diagonal diffusion matrix [23], where the maindifficulty consists in a lack of control of the reaction rates. Extensions to a cross-diffusionsystem from population dynamics with weak cross diffusion can be found in [12]. Bothreferences are based on the relative entropy method, but their arguments rely on thespecific structure of the diffusion matrix of their systems. Here, we would like to presenta more general strategy to deduce stability from an underlying thermodynamic structure.Given the strong coupling and lack of a priori bounds in L ∞ , there are several difficultiesin our ERDS that require an adaptation of the classical relative entropy approach to weak-strong stability. First, due to the lack of growth restrictions on the reactions and in somecases even the flux term (see the models in [24]), the evolution of the classical relativeentropy used to measure the distance between a renormalised solution z and a strongsolution ˜ z , cannot be properly controlled. This is due to the term − ˆ Ω D i h (˜ z ) dd t z i d x = ˆ Ω ∇ D i h (˜ z ) · ( A ij ( z ) ∇ z j ) d x − ˆ Ω D i h (˜ z ) R i ( z ) d x (1.6)arising in the formal computation of the time derivative of H rel ( z, ˜ z ). In fact, the availablea priori estimates do not ensure that A ij ( z ) ∇ z j ∈ L (Ω) and R i ( z ) ∈ L (Ω) for a.e. time.At the same time, the corresponding integrands in (1.6) do not have a sign, and there isno hope for the uncontrolled parts to cancel with some of the remaining terms appearingin dd t H rel ( z, ˜ z ). It is therefore necessary to adjust the relative entropy H rel ( z, ˜ z ). This issuehas already been encountered in [23]; it can be resolved by introducing a suitable smoothand compactly supported truncation function ξ ∗ = ξ ∗ ( z ) with ξ ∗ ( z ) = 1 if P ni =0 z i ≤ E for some E ≫ P ni =0 ˜ z i (see Section 3.1 for details) in the formula for the relative entropyvia H ∗ rel ( z, ˜ z ) := H ( z ) − ˆ Ω D i h (˜ z )( ξ ∗ ( z ) z i − ˜ z i ) d x − H (˜ z ) . K. HOPF
The relative entropy density adjusted in this fashion allows to remove the issue pointedout above. (Strictly speaking, in the term D h (˜ z )( z − ˜ z ) the truncation function isnot needed in the models considered in this paper, and for other applications it may behelpful to use a different choice such as D i h (˜ z )( ξ ∗ i ( z ) z i − ˜ z i ) with ξ ∗ i ( z ) ≡ i = 0, orversions thereof.)A second difficulty arising in the case of ERDS is the inherent coupling between concen-trations and energy density, which manifests itself in the circumstance that the entropydensity cannot be additively decomposed into terms depending only on an individualcomponent z i . This in turn leads to a non-diagonal diffusion matrix A ( z ) and rendersestimating the evolution of H ∗ rel ( z, ˜ z ) substantially more delicate than in the diagonalcase. One of the main contributions of this manuscript is to show that such estimatescan be achieved, with relatively little technical effort, by carefully exploiting the entropystructure.The energy component u plays a distinguished role in ERDS that has to be takenadvantage of when interested in a general analysis. At a technical level, the physicalconstraint of the convex function h ( u, c ) being non-increasing in u (to ensure a non-negative temperature) restricts the range of relevant functions ˆ σ to sublinearities such asˆ σ ( u ) = u ν for some ν ∈ [0 ,
1) (with ν = 0 corresponding to log). Unless ˆ σ ( u ) has closeto linear growth for large values of u , even the possibility of an existence theory solelybased on the entropy estimate is questionable in general dimensions. We are interested incovering more degenerate choices of ˆ σ , and therefore cannot purely rely on the (adjusted)relative entropy to measure the distance of a weak to the strong solution. Instead, weexploit the absence of source terms in the u -component of the evolution system, whichallows to give an extra, scalar-like structure to the evolution law for u . Here, we contentourselves with the arguably simplest choice of an L -structure, meaning that we considerweighted generalised distances of the formDist ∗ α ( z, ˜ z ) = H ∗ rel ( z, ˜ z ) + α k u − ˜ u k L (Ω) , α ∈ (0 , ∞ ) . This is consistent with the approach in [24] and allows us in particular to show the weak-strong uniqueness property for the potentially pathological solutions constructed in [24,Theorem 1.8], where a cross-diffusion dominant regime was considered with gradients ofthe internal energy density inducing a (possibly) non-integrable concentration flux. Fur-thermore, by exploiting the existence of such an additional quantity that up to some errorterm is dissipated along the flow, we can relax the conditions on the entropy functionalin [39] required for proving exponential convergence to equilibrium.1.3.
Technique.
Here, we briefly outline, at a formal level, the main points of our argu-ment showing a weak-strong stability estimate of the form dd t Dist ∗ α ( z, ˜ z ) . T,α,ξ ∗ Dist ∗ α ( z, ˜ z ) , (1.7)on any finite time horizon (0 , T ), T < T ∗ , where z is assumed to be a ‘weak’ (renormalised)solution and ˜ z a ‘strong’ solution of ( erds ) in (0 , T ∗ ) × Ω for some T ∗ ∈ (0 , ∞ ].First, letting dist ∗ α ( z, ˜ z ) = h ∗ rel ( z, ˜ z ) + α | u − ˜ u | , where h ∗ rel ( z, ˜ z ) = h ( z ) − D i h (˜ z )( ξ ∗ ( z ) z i − ˜ z i ) − h (˜ z ) , (1.8)we can write Dist ∗ α ( z, ˜ z ) = ´ Ω dist ∗ α ( z, ˜ z ) d x . We further recall that the function ξ ∗ = ξ ∗ ( z )will be chosen such that ξ ∗ ( z ) = 1 if P ni =0 z i ≤ E for an auxiliary parameter E ≫ P ni =0 ˜ z i .Then, if E = E (˜ z, min { α, } ) is chosen large enough, dist ∗ α ( z, ˜ z ) ≥ z ∈ [0 , ∞ ) n with equality if and only if z = ˜ z .To sketch the argument leading to (1.7), let us for simplicity only consider the casewhere A j ( z ) = a ( z ) δ j with a &
1. In this case, it will suffice to take α ≥
1. We
EAK-STRONG UNIQUENESS FOR ERDS 7 now assume that z and ˜ z are sufficiently regular solutions of ( erds ) (with A = M D h , M ≥ D i hR i ≤ z be such that k ˜ z k C , ([0 ,T ] × ¯Ω) < ∞ andinf (0 ,T ) × Ω ˜ z i > i ∈ { , . . . , n } and all T < T ∗ . To estimate the time evolution ofDist ∗ α ( z, ˜ z ), one formally computes dd t Dist ∗ α ( z, ˜ z ) = ˆ Ω ρ ( h ) d x + α ˆ Ω ρ ( g ) d x, where (see Lemma 3.4) ρ ( h ) := −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) (1.9)+ ∇ ( D i ( ξ ∗ ( z ) z j ) D j h (˜ z )) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:16) D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) (cid:17) · M il (˜ z ) ∇ D l h (˜ z ) − D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) R i (˜ z )+ ( D i h ( z ) − D j h (˜ z ) D i ( ξ ∗ ( z ) z j )) R i ( z ) , and ρ ( g ) := − a ( z ) |∇ u | − a (˜ z ) |∇ ˜ u | + a ( z ) ∇ u · ∇ ˜ u + a (˜ z ) ∇ u · ∇ ˜ u = − a ( z ) |∇ u − ∇ ˜ u | − ( a ( z ) − a (˜ z ))( ∇ u − ∇ ˜ u ) · ∇ ˜ u. (1.10)Thus, to show (1.7) it suffices to obtain a pointwise upper bound of the form ρ α := ρ ( h ) + αρ ( g ) . dist ∗ α ( z, ˜ z ) . (1.11)This pointwise estimate will be proved by distinguishing four cases determined by thevalue of the weak solution z = z ( t, x ) ∈ [0 , ∞ ) n at any given point ( t, x ). This casedistinction is motivated by the following observations:First, if z ∈ [0 , ∞ ) n with P ni =0 z i ≤ E for E = E (˜ z ) large enough, we want h ∗ rel ( z, ˜ z )to coincide with the classical relative entropy density h rel ( z, ˜ z ) = h ( z ) − D i h (˜ z )( z i − ˜ z i ) − h (˜ z ) to be able to use its distance-like properties. This will be ensured by choosing ξ ∗ ( z ) = 1 with D k ξ ∗ ( z ) = 0 for all k ∈ N + whenever P ni =0 z i ≤ E (cf. the definition of ξ ∗ in Sect. 3.1). Thus, if | z − ˜ z | is close to zero, the strict convexity, non-negativity andvanishing in z = ˜ z of dist ∗ α ( · , ˜ z ) imply that dist ∗ α ( z, ˜ z ) ∼ k ˜ z k L ∞ ,E | z − ˜ z | for | z | ≤ E . Inthis case, to show that ρ α is quadratically small in | z − ˜ z | , we write (using P i z i ≤ E ) ρ ( h ) = −∇ ( D i h ( z ) − D i h (˜ z )) · M il ( z ) ∇ ( D l h ( z ) − D l h (˜ z )) − ∇ ( D i h ( z ) − D i h (˜ z )) · ( M il ( z ) − M il (˜ z )) ∇ D l h (˜ z ) − ∇ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) · M il (˜ z ) ∇ D l h (˜ z )+ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) R i (˜ z )+ ( D i h ( z ) − D i h (˜ z ))( R i ( z ) − R i (˜ z )) , see case A + in the proof of Theorem 2.8 for details. In order to deal with the termsinvolving a gradient of z that appear in the second and the third term on the RHS, onewould like to exploit the non-positive first term on the RHS. Typically (such as in theERDS models considered in [24]), the submatrix ( M il ( z )) i,l =1 ,...,n will, however, degenerateas soon as c i ց i ∈ { , . . . , n } . Yet if min { z , . . . , z n } ≥ ι for some ι >
0, thenit is possible to assume that ( M il ( z )) i,l =1 ,...,n & ι I n . This, combined with the second linein (1.10) and suitable smoothness assumptions on M and h , will allow us to infer that ρ α . E,ι, ˜ z | z − ˜ z | whenever z ( t, x ) ∈ A + := { z ′ ∈ [0 , ∞ ) n : min { z ′ , . . . , z ′ n } ≥ ι, P ni =0 z i ≤ E } for some ι > E ≥ K. HOPF
To deal with the case z ( t, x ) ∈ A := { z ′ : min { z ′ , . . . , z ′ n } < ι, P ni =0 z i ≤ E } , we fix ι = ι (˜ z ) > z i ≥ ι for all i = 0 , . . . , n . This implies that | z − ˜ z | ≥ ι whenever z ∈ A . Thus, since | z − ˜ z | & ∗ α ( z, ˜ z ) (see Prop. 3.2). It then suffices to have suitable coercivity estimates on P ( z )that allow to absorb those terms on the RHS of (1.9) that involve gradients of z and donot have a sign by the first term on the RHS, which equals − P ( z ). (Such coercivityestimates are typically already needed in the construction of solutions.)It remains to consider the case P i z i > E , where E will be chosen large enough, inparticular such that E ≥ k P i ˜ z i k L ∞ ((0 ,T ) × Ω) + 1 and E ≥ E with E being such thatdist ∗ α ( z, ˜ z ) & P i c i log + ( c i )+ u +1 for all ˜ E ≥ E . If z supp ξ ∗ , ρ ( h ) takes a simple form.The entropy dissipating property of diffusion and reactions, and the Lipschitz regularityof the strong solution ˜ z are sufficient to deduce (1.11) in this case (referred to as z ∈ C ).The intermediate case (below referred to as case z ∈ B ), where 0 < ξ ∗ <
1, is somewhatmore delicate as can be seen in formula (1.9), where the second term on the RHS involvesproducts of gradients of the weak solution. In order to be able to absorb this term bythe first term on the RHS of (1.9), another scale E ′ ≫ E is introduced (for conveniencewe choose E ′ = E N , as in [23]), and ξ ∗ will be taken such that ξ ∗ ( z ) = 0 if and only if P ni =0 z i ≥ E ′ , ξ ∗ ( z ) = 1 if and only if P ni =0 z i ≤ E . On these scales, derivatives of ξ ∗ typically have an additional decay property enabling the desired absorption if E ′ E is largeenough.1.4. Outline.
The rest of the article is structured as follows. In Section 2 we intro-duce relevant definitions and hypotheses, and formulate our main results: a weak-stronguniqueness principle for renormalised solutions to ( erds ) (see Thm 2.8), a strong en-tropy dissipation inequality as used in the proof of Theorem 2.8 (see Prop. 2.9), and aresult on the exponential convergence to equilibrium (see Prop. 2.10). In Section 2.3 wepresent selected examples that our main results apply to, including the class of ERDSconsidered in [24] as well as a class of models with cross diffusion between species. Theweak-strong uniqueness principle is proved in Section 3, starting with several auxiliaryresults with the actual proof of Theorem 2.8 being given in Section 3.4. The proofs of theentropy dissipation inequality and the exponential convergence to equilibrium are givenin Sections 4 and 5 respectively. Some auxiliary results are gathered in Appendix 6.
Notations. • Summation convention : any unspecified summations of the form P i are to be under-stood as P ni =0 . For brevity, we use a summation convention for summing over thesystem’s components i = 0 , . . . , n in case of repeatedly occurring indices while omit-ting the summation symbol. In ambiguous situations the summation symbol will beused. Summations restricted to i = 1 , . . . , n (excluding the u -component) will alwaysbe made explicit. In our convention, summation over repeated indices has priority overother mathematical operations such as integration or taking the absolute value. Forinstance, by default we let | A ik ( z ) ∇ z k | = | P nk =0 A ik ( z ) ∇ z k | . • For technical concerns regarding the notation M il ( z ) ∇ D l h ( z ), we refer to Remark 2.3. • We denote by R = (0 , R , . . . , R n ) T the vectors of reaction rates. • Given T ∗ ∈ (0 , ∞ ], we let I = [0 , T ∗ ) denote the time horizon of interest. For T > T := (0 , T ) × Ω. • For functions f = f ( z , . . . , z n ) we let D i f = ∂f∂z i and D ij f = ∂ f∂z i ∂z j for i, j ∈ { , . . . , n } . • In estimates,
C < ∞ typically denotes a finite (sufficiently large) constant that maychange from line to line, while we often use ǫ > EAK-STRONG UNIQUENESS FOR ERDS 9 • For quantities
A, B ≥ A . B if there exists a fixed constant C < ∞ suchthat A ≤ CB . The notation A & B means B . A , while A ∼ B is to be understoodas both A . B and A & B being satisfied. In order to indicate dependencies of theconstant C = C ( p , . . . , p k ) on certain parameters p , . . . , p k , we write A . p ,...,p k B ,and analogously for & and ∼ . • Any dependence of constants and estimates on the regular solution ˜ z ∈ C , will usuallynot be explicitly indicated. • We let min( z ) := min { z , z , . . . , z n } for z = ( z , . . . , z n ) ∈ [0 , ∞ ) n . • By default, | · | denotes the Euclidean norm, e.g. | z | = ( P i | z i | ) • | z | = P ni =0 z i and | c | = P ni =1 c i . • For time-dependent integral functionals ´ Ω f (( t, x ) , z ( t, x )) d x , where z = z ( t, x ) denotesa ‘weak’ solution of ( erds ) taking in a suitable sense the data z in , we use the convention ˆ Ω f (( t, x ) , z ( t, x )) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 := ˆ Ω f (( T, x ) , z ( T, x )) d x − ˆ Ω f ((0 , x ) , z in ( x )) d x provided the terms on the RHS are well-defined. • For an open set U ⊂ R N , C k ( U ) denotes the space of continuous functions on U thatare continuously differentiable up to order k ∈ N . By C k,ν ( U ) = C k,ν loc ( U ), we denotethe space of functions in C k ( U ), whose k -th derivative is ν -H¨older continuous for some ν ∈ (0 ,
1] on compact subsets K ⊆ U . (We use the symbol C k,ν loc ( U ) for clarity’s sake.) Main results
Assumptions.
Throughout these notes, we let d ≥ ⊂ R d be a boundedLipschitz domain with | Ω | = 1. We further let T ∗ ∈ (0 , ∞ ] and I = [0 , T ∗ ).To prepare for stating our main result, the weak-strong uniqueness principle (The-orem 2.8), we gather the following conditions.(A1) Entropy: h ∈ C ((0 , ∞ ) n ) is of the form (h1), where ˆ σ and w i are supposed tosatisfy (h2).(A2) Reactions: R = (0 , R , . . . , R n ) ∈ C ([0 , ∞ ) n ) n (i) satisfy P ni =1 D i h ( z ) R i ( z ) ≤ , ∞ ) n (ii) are locally Lipschitz continuous in (0 , ∞ ) n .(A3) Mobility matrix: M ∈ (cid:0) C , ((0 , ∞ ) n ) ∩ C ([0 , ∞ ) n ) (cid:1) (1+ n ) × (1+ n ) and there existnon-negative functions m, a ∈ C , ((0 , ∞ ) n ) and ̟ ∈ { , } with 0 ≤ m . ̟ and a & M l = e M l + δ l m for l = 0 , . . . , n, (A3.a)for suitable e M l satisfying P nl =0 e M l D lj h = δ j a. Moreover, for all z ∈ [0 , ∞ ) n with min i z i ≥ ι for some ι > ǫ ( ι ) > M ( z ) ≥ diag( m ( z ) , , . . . ,
0) + ǫ ( ι ) diag(0 , , . . . , . (A3.b)By continuity, when ι = 0, (A3.b) holds true with ǫ ( ι ) = 0.Using our standard notation A ( z ) = M ( z ) D h ( z ), hp. (A3.a) implies that, formally, n X j =0 A j ( z ) ∇ z j = n X l =0 M l ( z ) ∇ D l h ( z ) = a ( z ) ∇ u + m ( z ) ∇ D h ( z ) . (A3.c)We further need certain bounds on the flux and the concentration gradients in termsof the entropy dissipation. For this purpose we define for non-negative functions z j ∈ L ( I, L (Ω)) such that ∇ ( z sj ) ∈ L ( I ; L (Ω)) for some s ∈ { , } for each j ∈ { , . . . , n } ,the quantity P ( z ) : = ∇ z : ( D h ( z ) A ( z ) ∇ z )= ∇ D i h ( z ) · ( M il ( z ) ∇ D l h ( z )) , where the second equality is to be understood in a formal sense, see Remark 2.3. By thepositive semi-definiteness of M imposed by hp. (A3.b), we have P ( z ) ≥ z , and more specifically, P ( z ) ≥ m ( z ) |∇ D h ( z ) | .(A4) For all K ≥ χ {| z |≤ K } |∇ c | . K p P ( z ) , (A4.a) χ {| z |≤ K } | n X j =0 A ij ( z ) ∇ z j | . K p P ( z ) (A4.b)for all i = 0 , . . . , n .(A5) For a ( z ) as in (A3), | a ( z ) ∇ u | . (1 + u ) p P ( z ) . Additionally, we often impose the following bound:(A6) For all 0 ≤ i ≤ nχ {| z | ≥ } |∇ z || n X j =0 A ij ( z ) ∇ z j | . | z | P ( z ) + | z ||∇ u | . If (A6) is not satisfied, we have to assume that ̟ = 0 in (A3) together with the condition:(A6’) For all 0 ≤ i ≤ n and all u ∈ (0 , χ { u ≥ u } χ {| z | ≥ } |∇ z || n X j =0 A ij ( z ) ∇ z j | . | z | P ( z ) + C ( | z | , u ) |∇ u | . Let us observe that (A6) = ⇒ (A6’).A selection of relevant examples fulfilling the above hypotheses is provided in Sec-tion 2.3.2.2. Definitions and Results.
Throughout this text, we write A ( z ) := M ( z ) D h ( z ),where h takes the form (h1), (h2). We further recall our summation convention (seeNotations 1.5). Definition 2.1 (Renormalised solutions) . Let I = [0 , T ∗ ) and suppose that the vector-valued function z = ( u, c , . . . , c n ) has non-negative components z i ≥ satisfying √ z i ∈ L ( I ; H (Ω)) or z i ∈ L ( I ; H (Ω)) for all i = 0 , . . . , n . Further suppose that for all E ≥ χ {| z |≤ E } A ik ( z ) ∇ z k ∈ L ( I ; L (Ω)) , for every i ∈ { , . . . , n } .We call such z a renormalised solution of the energy-reaction-diffusion system ( erds ) in Ω T ∗ := (0 , T ∗ ) × Ω with initial data z in if for all ξ ∈ C ∞ ( R n ≥ ) with compactly supported EAK-STRONG UNIQUENESS FOR ERDS 11 derivative Dξ , all ψ ∈ C ∞ ( I × ¯Ω) and almost all T ∈ (0 , T ∗ ) ˆ Ω ξ ( z ( T, · )) ψ ( T, · ) d x − ˆ Ω ξ ( z in ) ψ (0 , · ) d x − ˆ T ˆ Ω ξ ( z ) ∂ t ψ d x d t = − ˆ T ˆ Ω D ij ξ ( z ) A ik ( z ) ∇ z k · ∇ z j ψ d x d t − ˆ T ˆ Ω D i ξ ( z ) A ik ( z ) ∇ z k · ∇ ψ d x d t + ˆ T ˆ Ω D i ξ ( z ) R i ( z ) ψ d x d t. (2.1) Remark 2.2.
By approximation, given a renormalised solution z , the equality (2.1) can beseen to hold true for a larger set of test functions ψ ∈ C ( I × ¯Ω) with ∂ t ψ ∈ L ( I ; L (Ω)) , ∇ ψ ∈ L ( I ; L (Ω)) , and for truncation functions ξ ∈ C ( R n ≥ ) with supp Dξ compact. Remark 2.3 (Notation) . Let z denote a renormalised solution of ( erds ) in the senseof Def. 2.1. To keep notation simple and better emphasise the entropy structure of thediffusive part, we will often use a ‘symbolic’ notation writing M il ( z ) ∇ D l h ( z ) insteadof A ik ( z ) ∇ z k , where as before the summation convention is used. Likewise, we write ∇ D i h ( z ) instead of the more precise notation D ij h ( z ) ∇ z j . The point here is that while,by hypothesis, the weak derivatives ∇ z j , j = 0 , . . . , n, are well-defined (in the standardSobolev/distributional sense), the function D i h ( z ) may not be weakly differentiable. Remark 2.4.
The following equation, equivalent to (2.1) , can be obtained by ‘reversingthe product rule’ for ∇ ˆ Ω ξ ( z ( T, · )) ψ ( T, · ) d x − ˆ Ω ξ ( z in ) ψ (0 , · ) d x − ˆ T ˆ Ω ξ ( z ) ∂ t ψ d x d t = − ˆ T ˆ Ω ∇ (cid:0) D i ξ ( z ) ψ (cid:1) · M il ( z ) ∇ D l ( z ) d x d t + ˆ T ˆ Ω D i ξ ( z ) R i ( z ) ψ d x d t, (2.2) where we recall our convention M il ( z ) ∇ D l h ( z ) := A ik ( z ) ∇ z k , see Remark 2.3. Notice that thanks to the hypothesis of Dξ being compactly supported, no growthrestrictions have to be imposed on the reaction term R ( z ) in (2.1), and none of theflux terms A ik ( z ) ∇ z k is necessarily required to be integrable in order for the integralsin (2.1) to converge. At the same time, this restrictive condition on the set of admissibletruncation functions ξ means that recovering a separate (e.g. weak) formulation of asingle component i ∈ { , . . . , n } of the system (assuming integrability of A i k ( z ) ∇ z k and R i ( z )) is not immediate unless all components of the flux and the reactions areknown to be integrable. Thus, consistent with the existence result for ( erds ) in [24], ourweak-strong uniqueness principle for renormalised solutions additionally assumes a weakformulation for the energy component. Definition 2.5 (Weak solutions of energy equation) . Let z = ( u, c ) be as in Def. 2.1.We say that u is a weak solution of the energy component ∂ t u = div( A j ( z ) ∇ z j ) in Ω T ∗ with no-flux boundary conditions and initial condition u in if A j ( z ) ∇ z j ∈ L ( I ; L (Ω)) and if for all ϕ ∈ C ( I × ¯Ω) and almost all T < T ∗ ˆ Ω u ( T, · ) ϕ ( T, · ) d x − ˆ Ω u in ϕ (0 , · ) d x − ˆ T ˆ Ω u∂ t ϕ d x d t = − ˆ T ˆ Ω A j ( z ) ∇ z j · ∇ ϕ d x d t. (2.3)By carefully using lower-semicontinuity type properties of the entropy and entropy dis-sipation, the existence proof of global-in-time weak and renormalised solutions to ERDS as provided in [24] typically allows to show that for almost all T ∈ (0 , ∞ ) the constructedsolutions satisfy the entropy dissipation inequality H ( z ( T )) − H ( z in ) ≤ − ˆ T ˆ Ω P ( z ) d x d t + ˆ T ˆ Ω R i ( z ) D i h ( z ) d x d t, ( ed )where we recall the notation P ( z ) := ∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) ≥
0. Observe that thanksto the non-negativity of P ( z ) and − R i ( z ) D i h ( z ) (to be assumed throughout), any func-tion z = ( u, c ) with u ∈ L ∞ t L x and well-defined, measurable P ( z ) that satisfies ( ed ) with h ( z in ) ∈ L (Ω) necessarily has the regularity P ( z ) ∈ L (Ω T ) and R i ( z ) D i h ( z ) ∈ L (Ω T ).(For the estimate on H ( z ( T )) required in this argument, see (6.7) in the appendix.)We further note that the energy equation ∂ t u = div( A j ( z ) ∇ z j ) is satisfied in the weaksense by the solutions constructed in ref. [24], that ∇ u ∈ L (Ω T ), and that the quantity G ( u ) = k u k L (Ω) satisfies (with an equality) G ( u ( T )) − G ( u in ) ≤ − ˆ T ˆ Ω a ( z ) |∇ u | d x d t − ˆ T ˆ Ω m ( z ) ∇ D h ( z ) · ∇ u d x d t ( ene )for almost all T ∈ (0 , ∞ ). (In [24], the case m ≡ erds ) under a set ofhypotheses that is motivated by and compatible with the models in [24] (see system (2.5)below). Before stating the theorem, we need to specify what we understand by a ‘strong’solution. Definition 2.6 (Weak solution) . Let I = [0 , T ∗ ) . We call a function z ∈ L ( I ; W , (Ω)) , z = ( u, c , . . . , c n ) with z i ≥ for all i , a weak solution of ( erds ) in Ω T ∗ with initial data z in if A ij ( z ) ∇ z j , R i ( z ) ∈ L ( I ; L (Ω)) for all i , and if for all ψ ∈ C ∞ ( I × ¯Ω) n andalmost all T ∈ (0 , T ∗ ) ˆ Ω z i ( T, · ) ψ i ( T, · ) d x − ˆ Ω z in i ψ i (0 , · ) d x − ˆ T ˆ Ω z i ∂ t ψ i d x d t = − ˆ T ˆ Ω A ij ( z ) ∇ z j · ∇ ψ i d x d t + ˆ T ˆ Ω R i ( z ) ψ i d x d t. (2.4) Definition 2.7 (‘Strong’ solution) . Let I = [0 , T ∗ ) . We call z = z ( t, x ) a strong solutionof system ( erds ) in Ω T ∗ with data z in if it is a weak solution in the sense of Definition 2.6,has the regularity z ∈ C , ( I × ¯Ω) and satisfies inf Ω T z i > for i = 0 , . . . , n and every T ∈ (0 , T ∗ ) . Theorem 2.8 (Weak-strong uniqueness) . Let T ∗ ∈ (0 , ∞ ] . Assume hp. (A1) – (A5) .Further suppose that (A6) holds true, or alternatively that ̟ = 0 and that (A6’) is fulfilled.Let z in = ( u in , c in ) ∈ L (Ω) n , z in i ≥ for all i , and h ( z in ) ∈ L (Ω) , u in ∈ L (Ω) . Let z = ( u, c ) be a renormalised solution of system ( erds ) in Ω T ∗ taking the initial data z in ,and let u be a weak solution of the energy equation in the sense of Definition 2.5. Furthersuppose that, for almost all T ∈ (0 , T ∗ ) , z satisfies the entropy inequality ( ed ) and theenergy inequality ( ene ) with ∇ u ∈ L (Ω T ) . If ˜ z is a strong solution in Ω T ∗ in the senseof Definition 2.7 taking the same initial data z in , then z = ˜ z a.e. in Ω T ∗ . The proof of Theorem 2.8 will be completed in Section 3.4. In Section 2.3 we provide alist of examples covered by this theorem, including a class of (isoenergetic) cross-diffusionsystems.As mentioned in the introduction, the proof of Theorem 2.8 is based on a weak-strongstability type estimate with a generalised distance involving a modified relative entropy
EAK-STRONG UNIQUENESS FOR ERDS 13 and an L -part for the energy component (cf. Sec. 1.3). The entropy inequality ( ed ) isa fundamental ingredient in the proof. As pointed out above, the solutions constructedin ref. [24] enjoy this estimate. The hypothesis in Theorem 2.8 that admissible weaksolutions satisfy some kind of entropy/energy dissipation inequality is not uncommon insystems with strong nonlinearities; similar assumptions are encountered in the context ofweak-strong uniqueness in fluid dynamics problems, see e.g. [45, 42] and references therein.Yet for many of the models we are interested in, the entropy dissipation inequality ( ed ) aswell as the energy (in)equality ( ene ) can be derived for general renormalised solutions z in the sense of Definition 2.1 (with the u -component satisfying a scalar problem as inDef. 2.5) if one assumes integrability of the quantities P ( z ) and a ( z ) |∇ u | . In order toillustrate the ideas, we provide a proof of inequality ( ed ) for one of the models in theref. [24] considering ERDS of the form˙ u = div (cid:0) a ( u, c ) ∇ u + m ( z ) ∇ D h ( z ) (cid:1) , (2.5a)˙ c i = div (cid:16) m i ( u, c ) ∇ log (cid:16) c i w i ( u ) (cid:17) + a ( u, c ) c i w ′ i ( u ) w i ( u ) ∇ u (cid:19) + R i ( u, c ) , (2.5b)where, as before, h ( u, c ) satisfies (h1), (h2), and where a ( z ) := π ( z ) γ ( z ), with γ ( u, c ) := (cid:0) ∂ u h − n X i =1 c i (cid:0) w ′ i w i (cid:1) (cid:1) = − ˆ σ ′′ ( u ) − n X i =1 c i w ′′ i ( u ) w i ( u ) > . (2.6)System (2.5) is obtained as a special case of ( erds ) by choosing M ( z ) := diag (cid:0) m, m , . . . , m n (cid:1) + π µ ⊗ µ (2.7)with non-negative functions m, m i , π ∈ C ([0 , ∞ ) n ) to be specified below, and µ =(1 , µ , . . . , µ n ) determined by µ i ( u, c ) := w ′ i ( u ) w i ( u ) c i for i ∈ { , . . . , n } . Throughout, the bound 0 ≤ p π ( u, c ) . u (2.8)and the mild regularity condition p π ( u, c ) w ′ i ( u ) ∈ C ([0 , ∞ ) n ) , i = 1 , . . . , n, will beimposed, the latter ensuring the continuity of M in [0 , ∞ ) n .The functions m i ( u, c ), i ∈ { , . . . , n } , are assumed to take the following form forcertain a i ∈ C ([0 , ∞ ) n ) and κ ,i , κ ,i ≥ m i ( u, c ) = c i a i ( u, c ) , where a i ( u, c ) ∼ κ ,i + κ ,i c i . (2.9) Hypotheses of Model (M0).
Model (M0) consists of the following conditions: • Entropy density h is given by (h1), (h2) with coefficient functions satisfying( w ′ i ) . − w ′′ i w i (2.10)for all i ∈ { , . . . , n } . • Reactions R i ∈ C ([0 , ∞ ) n ), i = 1 , . . . , n , satisfy (A2.i), where R ≡ • M is given by (2.7)–(2.9) with ◦ rank-one part: π ( z ) ∼ γ ( z ) , where γ is given by (2.6). ◦ diagonal part: 0 ≤ m ( z ) . ̟ for some ̟ ∈ { , } , and m i ( u, c ) given by (2.9)with κ ,i = 1 , κ ,i = 0 for all i ∈ { , . . . , n } . We recall that the evolution law ( erds ) associated with Model (M0) takes the form (2.5a),(2.5b). (Cf. Lemma 6.1 and [24] for details.) It is further easy to see that condition (2.8)is compatible with the choice π ( z ) ∼ γ ( z ) for any power law ˆ σ ( u ) = u ν , ν ∈ (0 ,
1) andˆ σ ( u ) = log( u ).Model (M0) generalises the special case M ( u, c ) = const · ( D h ( u, c )) − consideredin [28]. It allows for species-dependent diffusivities, and genuinely contains cross termsin this case. More precisely, for models with species-dependent diffusion coefficients,thermodynamical consistency always leads to cross-diffusion effects, since for a diagonaldiffusion matrix A = diag( . . . ) ∈ R (1+ n ) × (1+ n ) that is not a multiple of the identity matrixthe product M = A ( D h ) − is not symmetric.In the derivation of the entropy dissipation inequality ( ed ), we need to assume that m ( u, c ) n X l =1 c l . (1 + u ) , (2.11)and have to impose the following conditions mainly serving to avoid issues for small valuesof u close to zero: ( m ( u, c ) | ˆ σ ′′ ( u ) | . , − ˆ σ ′′ ( v ) . − ˆ σ ′′ ( u ) + 1 for all v ≥ u. (2.12) Proposition 2.9 (Strong entropy dissipation inequality) . Let T ∗ ∈ (0 , ∞ ] . Let the hy-potheses of Model (M0) hold, and assume locally ǫ -H¨older continuous reactions R ∈ C ǫ loc ([0 , ∞ ) n ) for some ǫ ∈ (0 , . In addition, assume the bounds (2.11) , (2.12) ,and suppose that w i ∈ C ([0 , ∞ )) for i = 1 , . . . , n . Let z in = ( u in , c in ) , z in i ≥ , and u in , ˆ σ − ( u in ) ∈ L (Ω) and c in i ∈ L log L (Ω) for all i . Let z = ( u, c ) have non-negativecomponents and suppose that u ∈ L ∞ loc ([0 , T ∗ ); L (Ω)) , P ( z ) ∈ L (Ω T ) , a ( z ) |∇ u | ∈ L (Ω T ) for all T < T ∗ . If z is a renormalised solution (in the sense of Def. 2.1) of system ( erds ) in Ω T ∗ withinitial data z in , then the strong entropy dissipation inequality is satisfied, i.e. H ( z ( t )) − H ( z ( s )) ≤ − ˆ ts ˆ Ω P ( z ) d x d τ + ˆ ts ˆ Ω R i ( z ) D i h ( z ) d x d τ ( ed .s) for a.e. ≤ s < t < T ∗ , and for s = 0 and a.e. t ∈ (0 , T ∗ ) . In particular, ineq. ( ed ) holds true for a.e. T ∈ (0 , T ∗ ) . For the proof of Proposition 2.9, see Section 4. Some comments on generalisations ofProposition 2.9 to other models are provided in Section 2.3.In our final main result, we illustrate that a version of the generalised distance canfurther be used to prove exponential convergence to equilibrium. Exponential convergencein relative entropy has been studied at a formal level in [39] by means of log-Sobolev typeinequalities leading to entropy-entropy dissipation estimates (see also [28]). In contrast tothe present approach, [39] solely relies on the relative entropy, which restricts the resultsto thermal parts σ close to linear; for instance, the choice σ ( u ) ∼ log u related to gasdynamics is only admitted in dimensions d ≤
2. For simplicity, in the following resultwe disregard the reaction term and refer to [39] for applications on mass action-type With the understanding that H ( z (0)) = H ( z in ). EAK-STRONG UNIQUENESS FOR ERDS 15 reactions. We will further assume the strong energy inequality i.e. G ( u ( t )) − G ( u ( s )) ≤ − ˆ ts ˆ Ω a ( z ) |∇ u | d x d τ − ˆ ts ˆ Ω m ( z ) ∇ D h ( z ) · ∇ u d x d τ ( ene .s)for a.e. 0 ≤ s < t < T ∗ , and for s = 0 and a.e. t ∈ (0 , T ∗ ). Proposition 2.10 (Exponential convergence to equilibrium) . Recall that | Ω | = 1 and letthe hypotheses of Model (M0) hold. Let z in = ( u in , c in ) have non-negative components with ˆ σ − ( u in ) ∈ L (Ω) , u in ∈ L (Ω) , c in i ∈ L log L . Let z = ( u, c ) with u ∈ L ∞ loc ([0 , ∞ ) , L (Ω)) and ∇ u ∈ L ([0 , ∞ ) , L (Ω)) be a global-in-time renormalised solution of ( erds ) with R ≡ , and suppose that ( ed .s) and ( ene .s) are satisfied (with T ∗ = ∞ ).Then ´ Ω z i ( t, x ) d x = ´ Ω z in i ( x ) d x =: ¯ z i for all i ∈ { , . . . , n } and a.a. t > , and thereexist constants α ∈ (0 , and λ = λ (¯ z, α, Ω) > such that for a.e. t > α ( z ( t ) , ¯ z ) ≤ e − λt Dist α ( z in , ¯ z ) , (2.13) where Dist α ( z, ¯ z ) = H rel ( z, ¯ z ) + αG rel ( u, ¯ u ) (cf. eq. (1.5) ). See Section 5 for the proof of Prop. 2.10. For non-trivial reactions obeying massaction kinetics, the steady state ¯ z associated with ( erds ) is determined by solving aconstrained minimisation problem for the convex entropy functional H ( u, c ) imposingenergy conservation E ( u, c ) = ´ Ω u ! = E and further linear constraints taking into ac-count possible conservation laws for the concentrations (see [28, 39]). Prop. 2.10 considersthe simplest case, where all species have a conserved mass. Extension to more generalreactions is usually achieved by means of suitable coercivity estimates for the dissipationterm − D i h ( z ) R i ( z ) ≥ (M0) allows for m = 0, leading to en-ergy flux induced by temperature gradients, a maximum principle for the internal energydensity (as is crucially used in [28]) is not available here. We further note that, with astandard Csisz´ar–Kullback–Pinsker inequality [15, 44], the bound (2.13) can be used todeduce exponential convergence to equilibrium in L (Ω) × ( L (Ω)) n . Remark 2.11.
Observe that the condition ( ed .s) in Prop 2.10 is satisfied under theadditional hypotheses on the coefficient functions imposed in Prop. 2.9. Under suitableregularity hypotheses, the strong energy inequality ( ene .s) (with an equality) can be provedsimilarly as in [24, Lemma 6.1] . Examples.
Below, we provide relevant applications of the weak-strong uniquenessresult.2.3.1.
Energy-reaction-diffusion systems.
The hypotheses of Theorem 2.8 are compatiblewith the class of ERDS introduced in [24]. In that work, the existence of generalisedsolutions (weak or renormalised) has been established for two classes of models, bothtaking the form (2.5) with m ≡ (M0) (see page 13) with m ≡
0. A briefverification of the model hypotheses of Theorem 2.8 for (M0) is provided in the appendix,see Lemma 6.2. The existence analysis for (M0) focuses on reactions obeying suitablegrowth conditions, in which case there are global-in-time weak solutions. However, renor-malised solutions can be constructed by adapting the proof of [24, Theorem 1.8], and inthis case no growth restrictions on | R ( z ) | are required. Conceptually, the construction ofrenormalised solutions for (M0) is simpler than in [24, Theorem 1.8], since the diffusiveflux is integrable and the strong convergence of ∇ u is not required in case of (M0) . The second class of models considered in [24] will here be referred to as (M1) . It againtakes the form (2.5) with m ≡
0, and assumes the following conditions: • Entropy density h is given by (h1), (h2). • Reactions R i ∈ C ([0 , ∞ ) n ), i = 1 , . . . , n , satisfy (A2.i), where R ≡ • M is given by (2.7)–(2.9) with m ( z ) ≡
0, and m i ( u, c ) given by (2.9), where κ ,i , κ ,i satisfy one of the following:(i) κ ,i = 1 and κ ,i = 0 for all i (ii) κ ,i ≥ κ ,i = 1 for all i • Moreover, ◦ π γ & γ as in (2.6)) ◦ w ′ i ( u ) . − w ′′ i ( u ) √ π ◦ √ π w ′ i w i . A ( z ) ∇ z , which may be non-integrable. Model (M1.i)satisfies conditions (A3.a) (with ̟ = 0), (A3.b), (A4), (A5) and (A6’) of Theorem 2.8.We refer to [24, Section 2.2] (notably the proof of Lemma 2.3 therein), where the ne-cessary estimates can be found. For verifying (A6’), one should also use the fact thatthe coefficient function a ( z ) = π ( z ) γ ( z ) satisfies the bound χ { u ≥ u } | a ( u, c ) | . C ( | z | , u )for any u >
0. Thus, under the extra smoothness assumptions ˆ σ, w i ∈ C ((0 , ∞ )) and R i , m, m i , π ∈ C , ((0 , ∞ ) n ), Theorem 2.8 is applicable. We caveat that verifying ( ed )as it was done in Prop. 2.9 for (M0) is much more delicate for Model (M1.i) due to thepossibility of strong cross diffusion caused by a non-integrable diffusion flux. Whether ornot (M1.i) admits an analogue of Prop. 2.9 is an open question.2.3.2. Reaction-cross-diffusion systems.
Our results further apply to (isoenergetic) pop-ulation models generalising the two-species system for pattern formation by Shigesada,Kawasaki, and Teramoto (SKT). Reduction to the isoenergetic case is achieved by choos-ing ̟ = 0 (see (A3)) and u ≡ u in to be spatially constant, which is consistent withthe evolution law for the energy density u if ̟ = 0. Then, the given energy density u = u in ∈ R + can be regarded as a fixed system’s parameter (in particular ∇ u = 0) andone can write h = h ( c ) , A = A ( c ) , R = R ( c ) , and P = P ( c ) etc.The generalised SKT system as considered in [9] states ∂ t c i = ∇ · ( A ij ( c ) ∇ c j ) + R i ( c ) , t > , x ∈ Ω , A ij ( c ) ∇ c j · ν, t > , x ∈ ∂ Ω , ( skt )where A ij ( c ) = δ ij p i ( c ) + c i ∂p i ∂c j ( c ) for i, j = 1 , . . . , n with p i ( c ) = a i + P nk =1 a ik c sk forsuitable a il ≥ s >
0. Under certain hypotheses, this system has a generalisedgradient structure with entropy density given by h ( c ) = P ni =1 π i λ s ( c i ) for constants π i > λ given by (1.3), λ s ( r ) = r s − srs − + 1 for s = 1. Under a weak cross-diffusion (wc)condition (see eq. (12) in [9]) or the detailed balance (db) condition π i a ij = π j a ji for all i, j ∈ { , . . . , n } together with a i > , a ii > i , the matrix M ( z ) = A ( z )( D h ( z )) − satisfies the non-degeneracy condition (A3.b), i.e. M ( c ) & ι I n whenever min { c , . . . , c n } ≥ ι , see the explicit estimate in [12, Lemma 2.1] for s = 1, and [9, Section 2] for the generalcase under certain extra hypotheses. Observe that the detailed balance condition is EAK-STRONG UNIQUENESS FOR ERDS 17 equivalent to the symmetry of the mobility matrix M = A ( D h ) − , and thus ensures thesymmetry of the diffusive part K diff of the Onsager operator.a) Linear transition rates s = 1. In this case, assuming (wc) or (db) with a i > , a ii >
0, one has P ( c ) & P ni =1 (cid:0) |∇ c i | + |∇√ c i | (cid:1) and | A ( c ) ∇ c | . | c ||∇ c | . Thus,assumptions (A1), (A3)–(A6) on the entropy density and the mobility matrix aresatisfied. (The more general weights π i > h ( c ) as opposed to the unit weightsin (A1) do not impact the analysis.) We leave it to reader to verify that, byadapting the proof of Prop. 2.9 (see also [23, Prop. 5]), the entropy dissipationinequality ( ed ) can be proved for renormalised solutions to this system, whenassuming the regularity c i , √ c i ∈ L ( I ; H (Ω)) for all i. (2.14)Observe that this regularity is essentially equivalent to the assumption in Prop. 2.9that P ( z ) ∈ L (Ω T ) for all T < T ∗ . Thus, for ( skt ) with s = 1 and reac-tions satisfying (A2), Theorem 2.8 yields the weak-strong uniqueness of renor-malised solutions of the regularity (2.14). A similar result has been obtainedpreviously in [12, Theorem 1]. Here, we should caveat that the regularity as-sumption c i ∈ L ( I ; H (Ω)) and √ c i ∈ L ( I ; H (Ω)) is also needed in the proofof [12, Theorem 1], is, however, incompletely stated in this theorem. Moreover,our result shows that hypothesis ‘(H2.iii)’ in [12] on the reactions can be dropped.b) Nonlinear transition rates: an adjustment of our hypotheses further allows totreat system ( skt ) with superlinear transition rates s ∈ (1 , ξ ∗ ( c ) in the definition of the modified relative entropy density h ∗ rel ( z, ˜ z )(given by (1.8)) our technique can still be applied. See Remark 3.1 for technicaldetails.For sublinear transition rates, s <
1, the problem becomes more delicate sincea direct analogue of condition (A6) is not available. When relying exclusivelyon entropy estimates for a priori bounds, even the construction of renormalisedsolutions, which to the author’s knowledge is currently only available in the caseof linear transition rates [11] (but likely to be extendable to the case s ∈ [1 , s ∈ (0 , Weak-strong uniqueness principle
In this section we will establish a stability estimate implying Theorem 2.8. Throughoutthis section, we therefore assume the hypotheses of Theorem 2.8. Before turning to theactual proof in Subsection 3.4, we gather some technical auxiliary results.3.1.
Truncation function ξ ∗ ( z ) . Let ι, B be fixed constants, 0 < ι < < B < ∞ , thatare such that for all i ∈ { , . . . , n } ι ≤ ˜ z i ≤ B. (3.1) Different regimes/case distinction.
We henceforth abbreviate | z | = P ni =0 z i . Forsufficiently large but fixed parameters E, N ≥ , ∞ ) n into the following three sets: A = { z ∈ [0 , ∞ ) n : | z | ≤ E } , B = { z ∈ [0 , ∞ ) n : E < | z | < E N } , C = { z ∈ [0 , ∞ ) n : | z | ≥ E N } . A motivation for this decomposition is given in the introduction (see Section 1.3). Letus also mention that in the proof of Theorem 2.8 the set A will be further decomposedinto A + and A (defined in (3.18)). The parameter E will always be supposed to satisfy E ≥ B . A finite number of further lower bounds on E will be imposed later on. Adjusted relative entropy.
We now define h ∗ rel ( z, ˜ z ) = h ( z ) − D i h (˜ z )( ξ ∗ ( z ) z i − ˜ z i ) − h (˜ z ) , where ξ ∗ = ξ ( E,N ) ∈ C ∞ ([0 , ∞ ) n ), 0 ≤ ξ ∗ ≤
1, is a truncation function subordinate tothe above case distinction enjoying the following properties:(t1) ξ ∗ ( z ) = 1 and D k ξ ∗ ( z ) = 0 for all k ∈ N + if z ∈ A ,(t2) ξ ∗ ( z ) = 0 and D k ξ ∗ ( z ) = 0 for all k ∈ N + if z ∈ C ,and(t3) | Dξ ∗ ( z ) | . N | z | , | D ξ ∗ ( z ) | . N ( | z | ) for all z ∈ [0 , ∞ ) n . A function ξ ∗ that has these properties can be obtained as follows: let ϑ ∈ C ∞ b ( R ) benon-increasing with ϑ ( r ) = 1 for r ≤ ϑ ( r ) = 0 for r ≥ ξ ∗ ( z ) = ξ ( E,N ) ( z ) = ϑ (cid:18) log( | z | ) − log( E )log( E N ) − log( E ) (cid:19) . It is elementary to check that this choice satisfies the above properties. For instance,note that | Dξ ( E,N ) ( z ) | . N | z | · E ) if N ≥ Remark 3.1 (Superlinear transition rates) . When dealing with (isoenergetic) reaction-cross-diffusion population systems with superlinear transition rates s ∈ (1 , for concen-trations c , . . . , c n (see Example b) in Sec. 2.3.2), the decay property (t3) is no longersufficient, and the above choice of ξ ∗ should be replaced by ξ ∗ ( c ) := ξ ( E,N,s ) ( c ) := ϑ (cid:18) log( ρ ss ( c )) − log( E s )log( E sN ) − log( E s ) (cid:19) = ϑ (cid:18) log( ρ s ( c )) − log( E )log( E N ) − log( E ) (cid:19) , (3.2) where ϑ is as before, and ρ s ( c ) := (cid:16) n X i =1 ( c i + δ ) s (cid:17) s with δ = δ ( s ) = 1 for s ∈ (1 , and δ ( s ) = 0 for s = 2 . The parameter δ ∈ { , } ensuresthe smoothness of ρ ss and thus of ξ ∗ .Introducing the regimes A ( s ) = { ρ s ( c ) ≤ E } , B ( s ) = { E < ρ s ( c ) < E N } , C ( s ) = { ρ s ( c ) ≥ E N } , properties (t1) and (t2) remain valid with A , C replaced by A ( s ) and C ( s ) , respectively. Derivatives of ξ ∗ given by (3.2) enjoy the following decay properties,specifically adapted to the problem at hand, | D i ξ ∗ ( c ) | . ( c i + δ ) s − N | c | ss , | D ik ξ ∗ ( c ) | . ( c i + δ ) s − ( c k + δ ) s − N | c | ss + ( c i + δ ) s − N | c | ss δ ik , where δ ik denotes the Kronecker delta. As will become clear in Section 3.4, if c ∈ A ( s ) ,one can simply follow the reasoning in the proof of Theorem 2.8 using the fact that ( skt ) satisfies (under reasonable hypotheses) a non-degeneracy condition analogous to (A3.b) .If c ∈ B ( s ) , we need an analogue of assumption (A6) to be able to absorb terms withouta good sign involving products of gradients of the renormalised solution by the entropy EAK-STRONG UNIQUENESS FOR ERDS 19 dissipation. Typical systems ( skt ) satisfy the coercivity bound (cf. [9] ) P ( c ) & n X i =1 |∇ c si | + n X i =1 |∇ c s/ i | . (3.3) Confining to systems ( skt ) enjoying this bound, a suitable generalisation of hp. (A6) (inthe isoenergetic case) that is satisfied by such systems is χ {| c | s ≥ } · (cid:0) c s − i + c s/ − i (cid:1) | A ij ( c ) ∇ c j | . | c | ss p P ( c ) . Observe that since s ∈ (1 , , the factor (cid:0) c s − i + c s/ − i (cid:1) in this condition can be equivalentlyreplaced by (cid:0) ( c i + δ ) s − + c s/ − i (cid:1) . Using the above model bounds and decay properties ofderivatives of ξ ∗ , one can verify the estimate χ {| c | s ≥ } |∇ D i ( ξ ∗ ( c ) c l ) || A ij ( c ) ∇ c j | . N P ( c ) , which allows to deal with the case c ∈ B ( s ) (cf. ineq. (3.24) in the proof of Thm 2.8).When verifying this bound, one uses the fact that |∇ c l | . |∇ c sl | + |∇ c s/ l | , which holdstrue since s ∈ (1 , .If c ∈ C ( s ) , one can follow the reasoning in the proof of Thm 2.8.Thus, for models with superlinear transition rates s ∈ (1 , that satisfy (3.3) , weak-strong uniqueness is obtained by adapting the proof of Theorem 2.8 as sketched above.Here, we also use the fact that thanks to the locally uniform convexity of the entropydensity associated with ( skt ) (see Sec. 2.3.2) an analogue of Prop. 3.2 is immediate. Coercivity properties of the generalised distance.
We henceforth let g ( s ) = s , g rel ( u, ˜ u ) := g ( u ) − g ′ (˜ u )( u − ˜ u ) − g (˜ u ) = | u − ˜ u | , (3.4)and define for α ∈ (0 , ∞ ) dist ∗ α ( z, ˜ z ) = h ∗ rel ( z, ˜ z ) + αg rel ( u, ˜ u ) . For the following assertion we recall that z ∈ A if and only if P ni =0 z i ≤ E . Proposition 3.2 (Coercivity properties) . Recall that h ( z ) = h ( u, c ) is given by (h1) , (h2) ,and that ˜ z satisfies (3.1) . For any E ∈ [1 , ∞ ) , we have | z − ˜ z | . E h ∗ rel ( z, ˜ z ) if z ∈ A . (3.5) There exists E = E (˜ z ) < ∞ such that for any E ≥ E and any α ≥ h ∗ rel ( z, ˜ z ) + αg rel ( u, ˜ u ) ≥ ǫ (cid:18) n X i =1 c i log + c i + u (cid:19) + 1 if z ∈ A c , (3.6) where ǫ > is a positive constant only depending on model parameters.For any α ∈ (0 , there exists E = E ( α, ˜ z ) < ∞ such that for any E ≥ Eh ∗ rel ( z, ˜ z ) + αg rel ( u, ˜ u ) ≥ ǫ (cid:18) n X i =1 c i log + c i + α u (cid:19) + 1 if z ∈ A c , (3.7) where ǫ > is a positive constant only depending on model parameters. Remark 3.3.
Note that as long as α ∈ [1 , ∞ ) , the lower bound E can be chosen inde-pendently of α . This property is essential for obtaining the stability estimate (1.7) in thecase where (A6) is not satisfied (including model (M1) in case (i), where κ ,i = 0 ; cf.Section 2.3). We recall that any dependence of estimates on ι and B (see (3.1)) will usually not be indicatedexplicitly. Proof of Prop. 3.2.
For the first assertion, we note that, as can be seen from the proofof [39, Prop. 2.1], the entropy density h is locally uniformly convex on [0 , ∞ ) n with D ij h ( z ) ≥ ǫ ( E ) δ ij if | z | ≤ E . Since, by construction, ξ ∗ ( z ) = 1 whenever z ∈ A , wethus infer h ∗ rel ( z, ˜ z ) = h ( z ) − D i h (˜ z )( z − ˜ z ) i − h (˜ z ) & E | z − ˜ z | . Let us now turn to assertions (3.6) and (3.7). Since ˜ u ≤ B , we have the bound | u − ˜ u | ≥ u χ { u ≥ B } . By Lemma 6.3, we further have for some ν ∈ [0 ,
1) and somepositive constant ǫ > h ( z ) ≥ ǫ n X i =1 c i log + c i − C (ˆ σ + ( u ) + u ν ) − C. Hence, h ∗ rel ( z, ˜ z ) + αg rel ( u, ˜ u ) ≥ ǫ n X i =1 c i log + c i + αu χ { u ≥ B } − C n X i =0 z i − C. (3.8)Inequality (3.6) is now immediate since P ni =1 c i log( c i ) + u dominates P n z i for | z | ≥ E whenever E is large enough. Observe that the lower bound on E can be chosenindependently of α ∈ [1 , ∞ ).Let now α ∈ (0 ,
1] be given. Inequality (3.8) shows that if E = E ( α ) is large enough,we obtain (3.7) if E ≥ E .3.3. Evolution inequality for the generalised distance.
We will abbreviate H ∗ rel ( z, ˜ z ) := ˆ Ω h ∗ rel ( z, ˜ z ) d x, G rel ( u, ˜ u ) := ˆ Ω g rel ( u, ˜ u ) d x. In this subsection, we exploit the evolution laws satisfied by the renormalised solution z and the strong solution ˜ z of Theorem 2.8 to derive an evolution inequality for thegeneralised distanceDist ∗ α ( z, ˜ z ) := ˆ Ω dist ∗ α ( z, ˜ z ) d x = H ∗ rel ( z, ˜ z ) + α G rel ( u, ˜ u ) , (3.9)where α ∈ (0 , ∞ ) is a suitably chosen weight (to be specified in Section 3.4).We recall that I := [0 , T ∗ ). Furthermore, we note that since ∂ t ˜ z ∈ L ∞ (Ω T ) for any T < T ∗ , we can integrate by parts with respect to time in the weak formulation (2.4)satisfied by the strong solution ˜ z to find ˆ T ˆ Ω ∂ t ˜ z i ψ i d x d t = − ˆ T ˆ Ω A ij (˜ z ) ∇ ˜ z j · ∇ ψ i d x d t + ˆ T ˆ Ω R i (˜ z ) ψ i d x d t. (3.10)By a density argument, one can see that eq. (3.10) holds true for all ψ ∈ L ( I ; W , (Ω)).We consider separately the two quantities H ∗ rel ( z, ˜ z ) and G rel ( u, ˜ u ) appearing in (3.9),beginning with the former. Lemma 3.4 (Evolution of the entropic part) . For a.e.
T < T ∗ one has H ∗ rel ( z, ˜ z ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ ˆ T ˆ Ω ρ ( h ) d x d t, EAK-STRONG UNIQUENESS FOR ERDS 21 where ρ ( h ) := −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z )+ ∇ ( D i ( ξ ∗ ( z ) z j ) D j h (˜ z )) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:16) D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) (cid:17) · M il (˜ z ) ∇ D l h (˜ z ) − D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) R i (˜ z )+ ( D i h ( z ) − D j h (˜ z ) D i ( ξ ∗ ( z ) z j )) R i ( z ) . (3.11) Proof of Lemma 3.4.
The subsequent observations apply to a.e.
T < T ∗ .We write H ∗ rel ( z, ˜ z ) = H ( z ) − ˆ Ω D i h (˜ z ) ξ ∗ ( z ) z i d x + ˆ Ω ( D i h (˜ z )˜ z i − h (˜ z )) d x. For the first term on the RHS we use the fact that, by hypothesis, z satisfies ( ed ), i.e. H ( z ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ − ˆ T ˆ Ω ∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) d x d t + ˆ T ˆ Ω D i h ( z ) R i ( z ) d x d t. For the second term, we want to use the fact that z satisfies the renormalised formula-tions (2.1) and (2.2) with the truncation function ξ ( z ) = ξ ∗ ( z ) z j and the test function ψ = D j h (˜ z ) ∈ W , ∞ (Ω T ). (For the admissibility of this choice, see Remark 2.2.) Insertingthese choices in eq. (2.2), we obtain − ˆ Ω D j h (˜ z ) ξ ∗ ( z ) z j d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 + ˆ T ˆ Ω ξ ∗ ( z ) z j dd t D j h (˜ z ) d x d t = ˆ T ˆ Ω ∇ ( D i ( ξ ∗ ( z ) z j ) D j h (˜ z )) · M il ( z ) ∇ D l h ( z ) d x d t − ˆ T ˆ Ω D i ( ξ ∗ ( z ) z j ) R i ( z ) D j h (˜ z ) d x d t. We next rewrite the second term on the LHS choosing in the weak form (3.10) for ˜ z thetest function ψ := D ij h (˜ z ) ξ ∗ ( z ) z j . This yields ˆ T ˆ Ω D ij h (˜ z ) ξ ∗ ( z ) z j ∂ t ˜ z i d x d t = − ˆ T ˆ Ω ∇ (cid:0) D ij h (˜ z ) ξ ∗ ( z ) z j ) · M il (˜ z ) ∇ D l h (˜ z ) d x d t + ˆ T ˆ Ω D ij h (˜ z ) ξ ∗ ( z ) z j R i (˜ z ) d x d t. Observe that since χ {| z |≤ E } ∇ z ∈ L ( I ; L (Ω)) for any E < ∞ , the function ψ := D ij h (˜ z ) ξ ∗ ( z ) z j ∈ L ( I ; H (Ω)) is indeed admissible in the weak equation (3.10) for ˜ z .We finally need to determine the evolution of the term ˆ Ω ( D i h (˜ z )˜ z i − h (˜ z )) d x. To this end, note that thanks to the regularity of ˜ z , ˆ Ω h (˜ z ) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = ˆ T ˆ Ω D i h (˜ z ) ∂ t ˜ z i d x d t, ˆ Ω D i h (˜ z )˜ z i d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = ˆ T ˆ Ω D i h (˜ z ) ∂ t ˜ z i d x d t + ˆ T ˆ Ω (cid:0) dd t D i h (˜ z ) (cid:1) ˜ z i d x d t. Subtracting the first from the second equality then yields ˆ Ω (cid:0) D i h (˜ z )˜ z i − h (˜ z ) (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = ˆ T ˆ Ω D ij h (˜ z )˜ z i ∂ t ˜ z j d x d t = ˆ T ˆ Ω D ij h (˜ z )˜ z j ∂ t ˜ z i d x d t = − ˆ T ˆ Ω ∇ (cid:0) D ij h (˜ z )˜ z j (cid:1) · M il (˜ z ) ∇ D l h (˜ z ) d x d t + ˆ T ˆ Ω D ij h (˜ z )˜ z j R i (˜ z ) d x d t, where in the second step we have used the symmetry of the Hessian of h .In combination, the above estimates yield the bound H ∗ rel ( z, ˜ z ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ − ˆ T ˆ Ω ∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) d x d t + ˆ T ˆ Ω D i h ( z ) R i ( z ) d x d t + ˆ T ˆ Ω ∇ ( D i ( ξ ∗ ( z ) z j ) D j h (˜ z )) · M il ( z ) ∇ D l h ( z ) d x d t − ˆ T ˆ Ω D i ( ξ ∗ ( z ) z j ) R i ( z ) D j h (˜ z ) d x d t + ˆ T ˆ Ω ∇ (cid:0) D ij h (˜ z ) ξ ∗ ( z ) z j ) · M il (˜ z ) ∇ D l h (˜ z ) d x d t − ˆ T ˆ Ω D ij h (˜ z ) ξ ∗ ( z ) z j R i (˜ z ) d x d t − ˆ T ˆ Ω ∇ (cid:0) D ij h (˜ z )˜ z j (cid:1) · M il (˜ z ) ∇ D l h (˜ z ) d x d t + ˆ T ˆ Ω D ij h (˜ z )˜ z j R i (˜ z ) d x d t. The asserted inequality is now obtained upon rearranging the integrals on the RHS.We next turn to the energetic part. We first note that equation (3.10) and the factthat R ≡ ˆ T ˆ Ω ∂ t ˜ uϕ d x d t = − ˆ T ˆ Ω A j (˜ z ) ∇ ˜ z j · ∇ ϕ d x d t (3.12)for all ϕ ∈ L ( I ; W , (Ω)). Lemma 3.5 (Evolution of the energetic part) . Recall the definition of g rel in (3.4) andthe notation G rel ( u, ˜ u ) := ´ Ω g rel ( u, ˜ u ) d x . For almost every T > , we have G rel ( u, ˜ u ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = ˆ T ˆ Ω ρ g d x d t, (3.13) EAK-STRONG UNIQUENESS FOR ERDS 23 where ρ ( g ) := − a ( z ) |∇ u − ∇ ˜ u | − ( a ( z ) − a (˜ z ))( ∇ u − ∇ ˜ u ) · ∇ ˜ u − m ( z )( ∇ u − ∇ ˜ u ) · ( ∇ D h ( z ) − ∇ D h (˜ z )) − ( m ( z ) − m (˜ z ))( ∇ u − ∇ ˜ u ) · ∇ D h (˜ z ) . (3.14) Proof.
We expand g rel ( u, ˜ u ) = u − u ˜ u + ˜ u .To deal with the first term on the RHS, we use the energy inequality ( ene ), i.e. theproperty that for a.e. T < T ∗ G ( u ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ − ˆ T ˆ Ω a ( z ) |∇ u | d x d t − ˆ T ˆ Ω m ( z ) ∇ D h ( z ) · ∇ u d x d t. To determine the time evolution of the term ´ Ω u ˜ u d x , we assert that the Lipschitzfunction ˜ u is admissible in the weak formulation (2.3) of the equation for u , thus yielding − ˆ Ω u ˜ u d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 + ˆ T ˆ Ω u∂ t ˜ u d x d t = ˆ T ˆ Ω (cid:0) a ( z ) ∇ u · ∇ ˜ u + m ( z ) ∇ D h ( z ) · ∇ ˜ u (cid:1) d x d t, where we used (A3.c). The admissibility of ˜ u can be shown as follows: first exploit theregularity properties of ∇ u ∈ L ( I ; L (Ω)), u ∈ L ∞ loc ( I ; L (Ω)), which hold true by hypo-thesis resp. follow from ( ene ) and the fact that P ( z ) ∈ L ( I ; L (Ω)). Assumption (A5),Gagliardo–Nirenberg interpolation applied to u and the estimate m |∇ D h ( z ) | ≤ √ m p P ( z ) . p P ( z )then imply improved integrability of the flux term, namely for some s = s ( d ) > A j ( z ) ∇ z j = a ( z ) ∇ u + m ( z ) ∇ D h ( z ) ∈ L s loc ( I ; L s (Ω)) . With these bounds one can now use an approximation argument to show that, underthe current hypotheses, eq. (2.3) can be extended in particular to Lipschitz functions ϕ ∈ C , ( I × ¯Ω).Finally, using the test function ϕ = u − ˜ u ∈ L ( I ; W , (Ω)) in the weak equation (3.12)for ˜ u gives − ˆ T ˆ Ω ∂ t ˜ u ( u − ˜ u ) d x d t = ˆ T ˆ Ω a (˜ z ) ∇ ˜ u · ∇ ( u − ˜ u ) d x d t. The asserted identity (3.13) is now obtained by adding up the above equations andrearranging appropriately the terms on the RHS.The evolution inequality for our generalised distance is an immediate consequence ofthe previous two propositions.
Corollary 3.6.
Let α ∈ (0 , ∞ ) . We have Dist ∗ α ( z, ˜ z ) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ ˆ T ˆ Ω ρ α d x d t, (3.15) where ρ α := ρ ( h ) + αρ ( g ) with ρ ( h ) , ρ ( g ) given by (3.11) resp. (3.14) . Stability estimate.
Proof of Theorem 2.8.
Since (A6) implies (A6’), it suffices to prove the assertion for thecase ‘ ̟ = 1 (and thus, by hp., (A6))’ and the case ‘ ̟ = 0 and (A6’)’, henceforth referredto as Case ̟ = 1 resp. Case ̟ = 0.We will show the following. • Case ̟ = 1 : if α ∈ (0 ,
1] is sufficiently small, and if E = E ( α ) and N = N ( E )are large enough, then for almost all T ∈ (0 , T ∗ )Dist ∗ α ( z ( t, · ) , ˜ z ( t, · )) (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 . E,N,α ˆ T Dist ∗ α ( z, ˜ z ) d t. (3.16) • Case ̟ = 0 : if E , N = N ( E ) and α ∈ [1 , ∞ ) are chosen large enough ( α possiblydepending on E, N ), then for a.a. T ∈ (0 , T ∗ ) ineq. (3.16) holds true.Once ineq. (3.16) has been established, we can invoke Gronwall’s inequality to infer thatfor a.e. T ∈ (0 , T ∗ ) Dist ∗ α ( z ( T, · ) , ˜ z ( T, · )) ≤ Dist ∗ α ( z in , f z in ) exp( kT ) , where k = k ( E, N, α ) > . The estimates will also depend on the fixed constant ι > z ). This dependency will only be indicatedoccasionally and for the sake of clarity.In view of inequality (3.15) it suffices to show the pointwise bound ρ α . E,N,α dist ∗ α ( z, ˜ z ) . (3.17)An elementary ingredient in the proof of this bound will be the coercivity properties ofdist ∗ α (see Prop. 3.2). We anticipate that referring to Prop. 3.2 will be the only instance,where the present proof makes use of the more specific form of the entropy density h ( u, c )assumed in (A1). Loosely speaking, besides the locally strict convexity ensuring (3.5),we will rely on a lower bound on the generalised distance of the form dist ∗ α ( z, ˜ z ) & u for | z | ≫ A , B and C introduced on page 17, where, owing tothe degeneracies of M ( z ) occurring when one of the concentrations vanishes, we furtherdecompose the set A into A + := { z ′ : min( z ′ ) ≥ ι } ∩ A and A := { z ′ : min( z ′ ) < ι } ∩ A , (3.18)where min( z ′ ) := min { z ′ , . . . , z ′ n } for z ′ = ( z ′ , . . . , z ′ n ) ∈ [0 , ∞ ) n . This decompositionfurther serves to avoid regularity issues of h as z i ց i ∈ { , . . . , n } .If z ∈ ( A + ) c , we will make use of the following equivalent formula for ρ ( g ) ρ ( g ) = − a ( z ) |∇ u | − a (˜ z ) |∇ ˜ u | + a ( z ) ∇ u · ∇ ˜ u + a (˜ z ) ∇ u · ∇ ˜ u − m ( z )( ∇ u − ∇ ˜ u ) · ∇ D h ( z )+ m (˜ z )( ∇ u − ∇ ˜ u ) · ∇ D h (˜ z ) . Since, by hypothesis, 0 ≤ m ( z ) . ̟ and a ( z ) &
1, this implies that ρ ( g ) ≤ − a ( z )2 |∇ u | + C | a ( z ) ∇ u | + Cm ( z ) ̟ |∇ D h ( z ) | + C. (3.19)Using this form in the case when z ∈ ( A + ) c , we can avoid for instance issues due to a ( z )becoming singular as u → EAK-STRONG UNIQUENESS FOR ERDS 25
Finally, note that z ∈ A implies ξ ∗ ( z ) = 1 and D k ξ ∗ ( z ) = 0 ∀ k ∈ N + , so that, if z ∈ A ,one has by formula (3.11) ρ ( h ) = −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z )+ ∇ D i h (˜ z ) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:16) D ij h (˜ z )( z j − ˜ z j ) (cid:17) · M il (˜ z ) ∇ D l h (˜ z ) d x d t − D ij h (˜ z )( z j − ˜ z j ) R i (˜ z )+ ( D i h ( z ) − D i h (˜ z )) R i ( z ) . (3.20)We are now ready to tackle the four cases.Case z ∈ A + : in this case we have the control ι ≤ z i ≤ E for all i ∈ { , . . . , n } , andwe need to show that ρ α . | z − ˜ z | . We therefore rewrite formula (3.20) as ρ ( h ) = −∇ ( D i h ( z ) − D i h (˜ z )) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:0) D ij h (˜ z )( z j − ˜ z j ) (cid:1) · M il (˜ z ) ∇ D l h (˜ z )+ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) R i (˜ z )+ ( D i h ( z ) − D i h (˜ z ))( R i ( z ) − R i (˜ z ))= −∇ ( D i h ( z ) − D i h (˜ z )) · M il ( z ) ∇ ( D l h ( z ) − D l h (˜ z )) − ∇ ( D i h ( z ) − D i h (˜ z )) · ( M il ( z ) − M il (˜ z )) ∇ D l h (˜ z ) − ∇ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) · M il (˜ z ) ∇ D l h (˜ z )+ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) R i (˜ z )+ ( D i h ( z ) − D i h (˜ z ))( R i ( z ) − R i (˜ z )) . (3.21)Since, by hp. (A3.b), M ( z ) ≥ diag( m ( z ) ̟, , . . . ,
0) + ǫ ( ι ) diag(0 , , . . . ,
1) for a suitableconstant ǫ ( ι ) >
0, we have ∇ ( D i h ( z ) − D i h (˜ z )) · M il ( z ) ∇ ( D l h ( z ) − D l h (˜ z )) ≥ m ( z ) ̟ |∇ D h ( z ) − ∇ D h (˜ z ) | + ǫ ( ι ) δ |∇ D c h ( z ) − ∇ D c h (˜ z ) | for any δ ∈ (0 , δ = δ ( α ) will eventually be chosen smallenough to be specified below. For i ∈ { , . . . , n } we observe that since ι ≤ z j ≤ E for all j ∈ { , . . . , n } , the triangle inequality yields |∇ D i h ( z ) − ∇ D i h (˜ z ) | ≥ |∇ log( c i ) − ∇ log(˜ c i ) | − |∇ log( w i ( u )) − ∇ log( w i (˜ u )) |≥ c i |∇ c i − ∇ ˜ c i | − (cid:12)(cid:12) c i − c i (cid:12)(cid:12) |∇ ˜ c i | − |∇ u − ∇ ˜ u | w ′ i ( u ) w i ( u ) − (cid:12)(cid:12) w ′ i ( u ) w i ( u ) − w ′ i (˜ u ) w i (˜ u ) (cid:12)(cid:12) |∇ ˜ u |≥ ǫ ( E, ι ) |∇ c − ∇ ˜ c | − C ( E, ι ) |∇ u − ∇ ˜ u | − C ( E, ι ) | z − ˜ z | , where ǫ ( E, ι ) > M , |∇ ( D i h ( z ) − D i h (˜ z )) · ( M il ( z ) − M il (˜ z )) ∇ D l h (˜ z ) | . E,ι |∇ Dh ( z ) − ∇ Dh (˜ z ) || z − ˜ z | . E,ι | z − ˜ z ||∇ z − ∇ ˜ z | + | z − ˜ z | . Before estimating the remaining terms, we compute for f ∈ C ((0 , ∞ ) n ) ∇ (cid:0) f ( z ) − f (˜ z ) − D j f (˜ z )( z j − ˜ z j ) (cid:1) = D j f ( z ) ∇ z j − D j f (˜ z ) ∇ ˜ z j − D j f (˜ z ) ∇ ( z j − ˜ z j ) − D jk f (˜ z )( z j − ˜ z j ) ∇ ˜ z k = ( D j f ( z ) − D j f (˜ z )) ∇ z j − D jk f (˜ z )( z j − ˜ z j ) ∇ ˜ z k = ( D j f ( z ) − D j f (˜ z )) ∇ ( z j − ˜ z j )+ [ D k f ( z ) − D k f (˜ z ) − D jk f (˜ z )( z j − ˜ z j )] ∇ ˜ z k and, using Taylor’s theorem, for k = 0 , . . . , n | D k f ( z ) − D k f (˜ z ) − D jk f (˜ z )( z j − ˜ z j ) | . E,ι,f | z − ˜ z | . Letting f ( z ) = D i h ( z ), we infer since h ∈ C ((0 , ∞ ) n ) (see hp. (A1)) that |∇ (cid:0) D i h ( z ) − D i h (˜ z ) − D ij h (˜ z )( z j − ˜ z j ) (cid:1) | . ι,E | z − ˜ z ||∇ z − ∇ ˜ z | + | z − ˜ z | . Using the previous bounds to estimate the RHS of (3.21), recalling also hp. (A2.ii),and applying Young’s inequality and an absorption argument, we thus infer for suitable ǫ ( ι, E ) > ρ ( h ) ≤ − ǫ ( ι, E ) δ |∇ c − ∇ ˜ c | − m ( z ) ̟ |∇ D h ( z ) − ∇ D h (˜ z ) | + C ( E, ι ) δ |∇ u − ∇ ˜ u | + C ( δ, E, ι ) | z − ˜ z | . On the other hand, using the fact that z a ( z ) is locally Lipschitz continuous in(0 , ∞ ) n , we deduce from eq. (3.14) for suitable ǫ > C < ∞ (independent of E, ι ) ρ ( g ) ≤ − ǫ |∇ u − ∇ ˜ u | + C m ( z ) ̟ |∇ D h ( z ) − ∇ D h (˜ z ) | + C ( E, ι ) | z − ˜ z | , where we used the fact that 0 ≤ m ( z ) . ̟ .If ̟ = 1, we choose α ∈ (0 ,
1] small enough such that αC ≤ δ = δ ( α, E, ι ) sufficiently small such that δC ( E, ι ) ≤ αǫ . We may then conclude that ρ α = ρ ( h ) + αρ ( g ) ≤ C ( α, E, ι ) | z − ˜ z | . Let us emphasise that the smallness condition of α is independent of E .If instead ̟ = 0, we choose for given α ∈ [1 , ∞ ) the parameter δ = δ ( α, E, ι ) smallenough such that δ C ( E, ι ) ≤ ǫ α , and obtain as before ρ α ≤ C ( α, E, ι ) | z − ˜ z | . Case z ∈ A : in this case, we have no lower bound on z i away from zero, but sincemin( z ) ≤ ι and min(˜ z ) ≥ ι , we know that | z − ˜ z | ≥ ι . By Prop. 3.2 (cf. (3.5)) it thussuffices to prove that ρ α . E ρ ( h ) ≤ −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z )+ ∇ D i h (˜ z ) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:16) D ij h (˜ z )( z j − ˜ z j ) (cid:17) · M il (˜ z ) ∇ D l h (˜ z )+ C ( E ) ≤ − P ( z ) + C ( E ) p P ( z ) + C |∇ u | + C ( E ) , In the case ̟ = 0, it suffices to restrict α to the range 1 ≤ α < ∞ . EAK-STRONG UNIQUENESS FOR ERDS 27 where the last step uses hp. (A4.a) and hp. (A4.b). Hence, ρ ( h ) ≤ − P ( z ) + C ( E ) |∇ u | + C ( E ) . Next, by ineq. (3.19) and hp. (A4.b), ρ ( g ) ≤ − a ( z )2 |∇ u | + C ( δ , E ) + δ P ( z ) + Cm ( z ) ̟ |∇ D h ( z ) | (3.22)for any δ > ̟ = 1, we let δ = 1 and use the estimate m ( z ) ̟ |∇ D h ( z ) | ≤ P ( z ), whichfollows from hp. (A3), to see that after possibly decreasing α ∈ (0 ,
1] we have ρ α ≤− α a ( z )2 |∇ u | + C ( E ) |∇ u | + C ( E ) ≤ − α a ( z )4 |∇ u | + C ( α, E ).If ̟ = 0, we choose δ = α ≤
1. Then ρ α ≤ − P ( z ) − α a ( z )2 |∇ u | + a ( z )4 |∇ u | + C ( α, E ) + P ( z ) . α,E . Case z ∈ B : in this case derivatives of ξ ∗ do in general not vanish, but we know that1 ≪ E < | z | < E N . We estimate ρ ( h ) = −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z )+ ∇ ( D i ( ξ ∗ ( z ) z j ) D j h (˜ z )) · M il ( z ) ∇ D l h ( z )+ ∇ (cid:16) D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) (cid:17) · M il (˜ z ) ∇ D l h (˜ z ) − D ij h (˜ z )( ξ ∗ ( z ) z j − ˜ z j ) R i (˜ z )+ D i h ( z ) R i ( z ) − D j h (˜ z ) D i ( ξ ∗ ( z ) z j ) R i ( z ) ≤ − P ( z )+ C |∇ D i ( ξ ∗ ( z ) z j ) · M il ( z ) ∇ D l h ( z ) | + C ( E, N ) p P ( z ) + C ( E, N ) |∇ u | (by hp. (A4.b) , (A4.a))+ C ( E, N ) (using hp. (A2.i)) ≤ − P ( z )+ C |∇ D i ( ξ ∗ ( z ) z j ) · M il ( z ) ∇ D l h ( z ) | + C ( E, N ) |∇ u | + C ( E, N ) . In order to estimate the term |∇ D i ( ξ ∗ ( z ) z j ) · M il ( z ) ∇ D l h ( z ) | , we observe that |∇ D i ( ξ ∗ ( z ) z j ) | . | Dξ ∗ ( z ) ||∇ z | + | D ξ ∗ ( z ) z ||∇ z | . N | z | |∇ z | (by (t3)) . (3.23)We first consider the case ̟ = 1. Then (A6) is at our disposal, which yields using (3.23) |∇ D i ( ξ ∗ ( z ) z j ) || M il ( z ) ∇ D l h ( z ) | . N | z | |∇ z || X j A ij ( z ) ∇ z j | . N P ( z ) + N |∇ u | . Thus, choosing N = N (min { , α } ) sufficiently large, we infer ρ ( h ) ≤ − P ( z ) + min { , α } a ( z )8 |∇ u | + C ( E, N, α ) . Next, simiarly as in (3.22), we estimate ρ ( g ) ≤ − a ( z )2 |∇ u | + C ( E, N ) + P ( z ) + C m ( z ) ̟ |∇ D h ( z ) | . Decreasing α ∈ (0 , αC m ( z ) ̟ |∇ D h ( z ) | ≤ P ( z ), weobtain ρ α . N,E . It remains to consider the case where ̟ = 0 and (A6’) are fulfilled. Since u := inf u in =inf ˜ u (0 , · ) ≥ ι , Lemma 6.4 yields inf u ≥ u >
0. By hp. (A6’), we infer for all 0 ≤ i ≤ n |∇ z || M il ( z ) ∇ D l h ( z ) | ≤ C | z | P ( z ) + C ( E, N, u ) |∇ u | . Thus, recalling ineq. (3.23), we can estimate for N large enough |∇ D i ( ξ ∗ ( z ) z j ) · M il ( z ) ∇ D l h ( z ) | ≤ P ( z ) + C ( E, N, u ) |∇ u | (3.24)to infer ρ ( h ) ≤ − P ( z ) + C ( E, N, u ) |∇ u | + C ( E, N ) . Next, since ̟ = 0, ineq. (3.19) yields ρ ( g ) ≤ − a ( z )2 |∇ u | + C ( E, N ) p P ( z ) + C ≤ − a ( z )2 |∇ u | + α P ( z ) + C ( α, E, N ) . Increasing α = α ( E, N, u ), if necessary, to ensure that α a ( z )2 ≥ C ( E, N, u ), we conclude ρ α ≤ − P ( z ) + C ( α, E, N, u ) . Case z ∈ C : in this case, ξ ∗ ( z ) = 0 and D k ξ ∗ ( z ) = 0 for all k ∈ N + . Thus ρ ( h ) = −∇ D i h ( z ) · M il ( z ) ∇ D l h ( z ) − ∇ ( D ij h (˜ z )˜ z j ) · M il (˜ z ) ∇ D l h (˜ z )+ D ij h (˜ z )˜ z j R i (˜ z )+ D i h ( z ) R i ( z ) ≤ − P ( z ) + C, where we used hp. (A2.i).If ̟ = 1, we have thanks to (3.19) and hp. (A3), (A5) ρ ( g ) ≤ − a ( z )2 |∇ u | + C P ( z ) + u + C, where C is independent of E, N . Hence, after possibly decreasing α ∈ (0 ,
1] to ensurethat αC ≤
1, we find ρ α ≤ C + αu . α dist ∗ α ( z, ˜ z ) , where the second step follows from (3.7) (after choosing E = E ( α ) large enough).If ̟ = 0, we estimate using again (3.19) and hp. (A5) ρ ( g ) ≤ − a ( z )2 |∇ u | + C (1 + u ) p P ( z ) + C ≤ − a ( z )2 |∇ u | + α P ( z ) + C ( α ) u + C ( α ) , and infer ρ α ≤ − P ( z ) − α a ( z )2 |∇ u | + C ( α ) u + C ( α ) . α dist ∗ α ( z, ˜ z ) . The second step follows from the coercivity property (3.6) and the fact that α ≥ EAK-STRONG UNIQUENESS FOR ERDS 29 Strong Entropy dissipation property
Proof of Proposition 2.9.
We first establish ( ed .s) for s = 0 and a.e. t = T ∈ (0 , T ∗ ), thatis, we first prove ( ed ). In a second step (see page 34 ), we point out how to extend theresult to a.e. 0 < s < t < T ∗ .Case 1: s = 0 , t = T ∈ (0 , T ∗ ).We consider for a small parameter δ > h δ ( u, c ) = δu + h ( u, c ) . The additive term δu serves to ensure coercivity, since the original density h ( z ) may ingeneral allow for cancellations at infinity reflecting the coupling between concentrationsand energy component. In particular, for every L ∈ N the sublevel set { z ∈ R n ≥ : h δ ( z ) ≤ L } is bounded. This coercivity property easily follows from the lower bound h ( z ) ≥ − ˆ σ ( u ) + ǫ ∗ n X i =1 c i log c i − Cu ν − C, (4.1)valid for suitable ν ∈ [0 , ǫ ∗ > σ ( u ) as u → ∞ (see (h2)).In order to define an admissible truncation function, we consider as in [23, Proof ofProp. 5] for L ≥ θ L ∈ C ∞ ( R ) satisfying θ L ( s ) = s for | s | ≤ L ,0 ≤ θ ′ L ≤ | θ ′′ L ( s ) | . C | s | log( | s | +e) (4.2)for all s ∈ R and θ ′ L ( s ) = 0 for | s | ≥ L C for some sufficiently large constant C ≥
2, whichis kept fixed throughout the proof.To derive the entropy dissipation inequality, we would like to choose the truncationfunction θ L ( h δ ( · )) and the test function ψ ≡ z , and then let L → ∞ and subsequently δ →
0. Since derivatives of h δ are in generalnot bounded as u ց c i ց
0, further regularisation is required. We let h δ,ε ( u, c ) = h δ ( z ) + ˆ σ ( u ) − ˆ σ ( u + ε ) , abbreviate for z = ( u, c , . . . , c n ) z ˜ ε := ( u, c + ˜ ε, . . . , c n + ˜ ε ) , ˜ ε ∈ (0 , , and then consider the function z h δ,ε ( z ˜ ε ) ∈ C ([0 , ∞ ) n ). Thanks to (4.1), it is easyto see that for fixed δ ∈ (0 ,
1] sublevel sets of z h δ,ε ( z ˜ ε ) are bounded; in fact h δ,ε ( z ˜ ε ) ≥ δ u + ǫ ∗ n X i =1 c i log + c i − C δ (4.3)for δ, ε, ˜ ε ∈ (0 , C -function ξ ( z ) := θ L ( h δ,ε ( z ˜ ε )) has compactly supportedderivative Dξ , and is thus an admissible truncation in eq. (2.1) (cf. Remark 2.2). This, combined with the choice ψ ≡ T < T ∗ LHS := ˆ Ω θ L ( h δ,ε ( z ˜ ε )) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 = − ˆ T ˆ Ω θ ′ L ( h δ,ε ( z ˜ ε )) D ij h δ,ε ( z ˜ ε ) ∇ z j · A ik ( z ) ∇ z k d x d t − ˆ T ˆ Ω θ ′′ L ( h δ,ε ( z ˜ ε )) D j h δ,ε ( z ˜ ε ) D i h δ,ε ( z ˜ ε ) ∇ z j · A ik ( z ) ∇ z k d x d t + ˆ T ˆ Ω θ ′ L ( h δ,ε ( z ˜ ε )) D i h ( z ˜ ε ) R i ( z ) d x d t = : I + II + III, (4.4)where we recall the summation convention (see Notations 1.5). In the reaction term wehave used the fact that D i h δ,ε ( z ) = D i h ( z ) whenever i = 0 together with R ≡ ed ) by taking the lim inf of the LHS andthe lim sup of the RHS of the above equation (4.4) as ˜ ε → L → ∞ and δ, ε →
0, inthe stated order. We perform the corresponding limits separately in
LHS and in each ofthe three terms
I, II, III . Below we use, without explicit reference, the following basicproperties satisfied under the hypotheses of Model (M0) , see Lemma 6.1: a ( z ) |∇ u | ∼ |∇ u | , P ( z ) & n X i =1 |∇√ c i | + |√ γ ∇ u | + |√ m ∇ D h ( z ) | , where γ ( u, c ) = − ˆ σ ′′ ( u ) − P nl =1 w ′′ l ( u ) w l ( u ) c l , | A ik ( z ) ∇ z k | . √ c i p P ( z ) for i ≥ , | A k ( z ) ∇ z k | . |∇ u | + √ m p P ( z ) . LHS:
The limit ˜ ε → LHS is immediate due to the boundedness of θ L , and yields ˆ Ω θ L ( h δ,ε ( z )) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 . We next take lim inf ε,δ →∞ lim inf L →∞ of the last expression using a combination of the dominated convergence theorem andFatou’s lemma:At initial time, we estimate using the lower and upper bounds on h in Lemma 6.3 | θ L ( h δ,ε ( z in )) | ≤ | h δ,ε ( z in ) | . u in + | ˆ σ − ( u in ) | + n X i =0 c in i log + c in i + 1 . We can hence use dominated convergence to deduce that, as L → ∞ and δ, ε → ˆ Ω θ L ( h δ,ε ( z in )) d x → ˆ Ω h ( z in ) d x. To deal with the integral at time t = T , we first observe that, thanks to the regularity u ∈ L ∞ loc ([0 , T ∗ ) , L (Ω)) and (4.1), the negative part of θ L ( h δ,ε ( z ( T, · ))) is controlled pointwisein x , for a.e. T ∈ (0 , T ∗ ], by an integrable function, uniformly in L, δ, ε , and its integral
EAK-STRONG UNIQUENESS FOR ERDS 31 can thus be shown to converge in the same way as the term at initial time. For thepositive part of θ L ( h δ,ε ( z ( T, · ))) we use Fatou’s lemma: ˆ Ω max { h δ,ε ( z ( T, · )) , } d x ≤ lim inf L →∞ ˆ Ω max { θ L ( h δ,ε ( z ( T, · ))) , } d x, ˆ Ω max { h ( z ( T, · )) , } d x ≤ lim inf δ,ε → ˆ Ω max { h δ,ε ( z ( T, · )) , } d x. In combination, we infer ˆ Ω h ( z ) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 ≤ lim inf δ,ε → lim inf L →∞ lim ˜ ε → ˆ Ω θ L ( h δ,ε ( z ˜ ε )) d x (cid:12)(cid:12)(cid:12)(cid:12) t = Tt =0 . Diffusive dissipation term I:
We assert thatlim sup δ,ε → lim sup L →∞ lim sup ˜ ε → (cid:16) − ˆ T ˆ Ω θ ′ L ( h δ,ε ( z ˜ ε )) ∇ z j · D ji h δ,ε ( z ˜ ε ) A ik ( z ) ∇ z k d x d t (cid:17) can be bounded above by the non-positive term − ˆ T ˆ Ω ∇ z j · D ji h ( z ) A ik ( z ) ∇ z k d x d t. To show this, we will mainly rely on the dominated convergence theorem. We thereforestart by listing several uniform pointwise estimates on the terms involved.We first estimate for i, j fixed the term p ij := θ ′ L ( h δ,ε ( z ˜ ε )) D ji h δ,ε ( z ˜ ε ) ∇ z j · X k A ik ( z ) ∇ z k , where we recall that 0 ≤ θ ′ L ≤ . Case i, j ≥ D ij h δ,ε ( z ˜ ε ) = c i +˜ ε δ ij and thus | p ij | ≤ | c i +˜ ε ∇ c i · X k A ik ( z ) ∇ z k | . c i |∇ c i |√ c i p P ( z ) . P ( z ) . Case i ≥ , j = 0 : observing that − D i h δ,ε ( z ˜ ε ) = w ′ i w i . q − w ′′ i w i (cf. (2.10)), we find | p ij | ≤ | w ′ i w i ∇ u · X k A ik ( z ) ∇ z k | . | q − w ′′ i w i c i ∇ u | p P ( z ) . P ( z ) . Case i = 0 , j ≥ k in the definition of p j into two parts: X k A k ( z ) ∇ z k = a ( z ) ∇ u + m ( z ) ∇ D h ( z )and split p j accordingly into p j = p (0)0 j + p (1)0 j . For j ≥ | p (0)0 j | ≤ | w ′ j w j ∇ c j · a ( z ) ∇ u | . | r − w ′′ j w j c j ∇ u ||∇√ c j | . P ( z ) . For j = 0, we have p (0)00 = θ ′ L ( h δ,ε ( z ˜ ε )) D h δ,ε ( z ˜ ε ) a ( z ) |∇ u | ≥
0, and since in the equationit comes with a minus sign, its integral can easily be handled using Fatou’s lemma.It remains to estimate the part p (1)0 j . To deal with the limit ˜ ε → L ), weestimate |∇ D h δ,ε ( z ˜ ε ) · m ( z ) ∇ D h ( z ) | . √ m |∇ D h δ,ε ( z ˜ ε ) | p P ( z ) and for | z ˜ ε | ≤ C ( L, δ ) √ m |∇ D h δ,ε ( z ˜ ε ) | ≤ | D h δ,ε ( z ˜ ε ) ∇ u | + n X j =1 | D j h δ,ε ( z ˜ ε ) ∇ c j | . L,δ,ε |∇ u | + p P ( z ) . Thus the limit ˜ ε → ε = 0 and compute ∇ D h δ,ε ( z ) · m ( z ) ∇ D h ( z )= m ( z ) |∇ D h ( z ) | − (ˆ σ ′′ ( u + ε ) − ˆ σ ′′ ( u )) ∇ u · m ( z ) ∇ D h ( z ) . Note that the first term on the RHS is bounded above by P ( z ) ∈ L (Ω T ). Concerningthe second term, we estimate | (ˆ σ ′′ ( u + ε ) − ˆ σ ′′ ( u )) ∇ u · m ( z ) ∇ D h ( z ) | . √ m | (ˆ σ ′′ ( u + ε ) − ˆ σ ′′ ( u )) ∇ u | p P ( z ) . | (1 + p − ˆ σ ′′ ( u )) ∇ u | p P ( z ) . P ( z ) + |∇ u | , where in the penultimate step we have used hypothesis (2.12).Combining the above estimates allows to take the successive limitslim sup ε,δ → lim sup L →∞ lim sup ˜ ε → . . . as above, thus yielding the asserted inequality for term I . Remainder gradient term II:
We will show that lim sup L →∞ lim sup ˜ ε → (cid:16) − ˆ T ˆ Ω θ ′′ L ( h δ,ε ( z ˜ ε )) D j h δ,ε ( z ˜ ε ) ∇ z j · D i h δ,ε ( z ˜ ε ) A ik ( z ) ∇ z k d x d t (cid:17) ≤ . (4.5) As in the previous paragraph, the main task is to obtain uniform pointwise estimates onthe terms involved, where here we can afford a dependence of our estimates on δ and ε .We introduce for i, j fixed the term q ij = − θ ′′ L ( h δ,ε ( z ˜ ε )) D j h δ,ε ( z ˜ ε ) ∇ z j · D i h δ,ε ( z ˜ ε ) X k A ik ( z ) ∇ z k and observe that, by (4.2) and (4.3), for L ≥ L ( δ ) (henceforth to be assumed) | θ ′′ L ( h δ,ε ( z ˜ ε )) | . δ u + P i c i log + ( c i )) log( | z | +e) . (4.6)Case i, j ≥ | q ij | . | θ ′′ L ( h δ,ε ( z ˜ ε )) || log (cid:16) c i +˜ εw i ( u ) (cid:17) || log (cid:16) c j +˜ εw j ( u ) (cid:17) |√ c i √ c j |∇√ c j | p P ( z ) . Estimating √ c i | log (cid:16) c i +˜ εw i ( u ) (cid:17) | . √ c i log + ( c i ) + p w i ( u ) log + ( w i ( u )) + 1 , we find, using (h2) and (4.6), | θ ′′ L ( h δ,ε ( z ˜ ε )) | √ c i | log (cid:16) c i +˜ εw i ( u ) (cid:17) | √ c j | log (cid:16) c j +˜ εw j ( u ) (cid:17) | . | θ ′′ L ( h δ,ε ( z ˜ ε )) | ( √ c i log + ( c i ) + √ u + 1)( √ c j log + ( c j ) + √ u + 1) . δ . Thus, | q ij | . δ P ( z ). EAK-STRONG UNIQUENESS FOR ERDS 33
Case i ≥ , j = 0 : here, | q ij | . √ c i | log (cid:16) c i +˜ εw i ( u ) (cid:17) || D h δ,ε ( z ˜ ε ) ∇ u | p P ( z ) | θ ′′ L ( h δ,ε ( z ˜ ε )) | . In view of the factor √ c i , this shows that | q ij | ≤ C ( L, δ, ε ) |∇ u | p P ( z ) uniformly in ˜ ε ,allowing us to infer by dominated convergencelim sup ˜ ε → ˆ T ˆ Ω q ij d x d t ≤ − ˆ T ˆ Ω θ ′′ L ( h δ,ε ( z )) D h δ,ε ( z ) ∇ u · log (cid:16) c i w i ( u ) (cid:17) A ik ( z ) ∇ z k d x d t. Since ε > w ′ l ( u ) w l ( u ) . q − w ′′ l ( u ) w l ( u ) , we have the rough bound | D h δ,ε ( z ) ∇ u | . ε |∇ u | + X l √ c l √ γ |∇ u | . |∇ u | + X l √ c l p P ( z ) . (4.7)Moreover, | log (cid:16) c i w i ( u ) (cid:17) A ik ( z ) ∇ z k | . (log + ( c i ) + 1 + log + ( w i ( u ))) √ c i p P ( z ) , and hence | θ ′′ L ( h δ,ε ( z )) D h δ,ε ( z ) ∇ u · D i h δ,ε ( z ) A ik ( z ) ∇ z k | . ε P ( z ) + |∇ u | . Case i = 0 , j ≥ | z | ≤ C ( L, δ ) | D j h δ,ε ( z ˜ ε ) ∇ z j || D h δ,ε ( z ˜ ε ) || a ( z ) ∇ u + m ( z ) ∇ D h ( z ) | . L,δ,ε P ( z ) + |∇ u | , one can take the limit ˜ ε → ˜ ε → (cid:0) − ˆ T ˆ Ω θ ′′ L ( h δ,ε ( z ˜ ε )) D j h δ,ε ( z ˜ ε ) ∇ z j · D h δ,ε ( z ˜ ε ) X k A k ( z ) ∇ z k d x d t (cid:1) ≤ − ˆ T ˆ Ω θ ′′ L ( h δ,ε ( z )) D j h δ,ε ( z ) ∇ z j · D h δ,ε ( z ) X k A k ( z ) ∇ z k d x d t. To obtain L -uniform bounds of the integrand on the RHS, we split X k A k ( z ) ∇ z k = a ( z ) ∇ u + m ( z ) ∇ D h ( z ) . Using (4.7) we have for j ≥ |∇ z j D j h δ,ε ( z ) || D h δ,ε ( z ) a ( z ) ∇ u | . ε √ c j (log + ( c j ) + 1 + log + ( w j ( u )))( |∇ u | p P ( z ) + X l √ c l P ( z )) , and for j = 0 |∇ uD h δ,ε ( z ) || D h δ,ε ( z ) a ( z ) ∇ u | . ǫ (cid:0) |∇ u | + p | c | p P ( z ) (cid:1) . |∇ u | + | c | P ( z ) . It remains to consider the term involving m ( z ). We estimate for j ≥ | D j h δ,ε ( z ) ∇ z j · D h δ,ε ( z ) || m ( z ) ∇ D h ( z ) | . ε √ c j (log + ( c j ) + 1 + log + ( w j ( u )))(1 + P l c l u ) √ m P ( z ) , where we used the fact that w ′ l ( u ) w l ( u ) . u for all l allowing us to estimate | D h δ,ε ( z ) | . ε P nl =1 c l u . Similarly, for j = 0 we estimate, using also the bound (4.7), | D h δ,ε ( z ) ∇ u · D h δ,ε ( z ) || m ( z ) ∇ D h ( z ) | . ε (1 + P nl ′ =1 c l ′ u ) (cid:0) |∇ u | + n X l =1 √ c l p P ( z ) (cid:1) √ m p P ( z ) . Thus, recalling the conditions (2.11), (2.12) on m ( z ), we infer the L -uniform bound | θ ′′ L ( h δ,ε ( z )) D j h δ,ε ( z ) ∇ z j · D h δ,ε ( z ) X k A k ( z ) ∇ z k | . δ,ε P ( z ) + |∇ u | . Combining the above estimates, we can take the limitslim sup L →∞ lim sup ˜ ε → . . . of term II and obtain ineq. (4.5) by the pointwise convergence θ ′′ L ( s ) → L → ∞ . Reactions III:
Concerning the reaction term
III , we first need to take care of the fact that D i h ( z ) isunbounded near c i = 0. By (4.3), we have | z | , | z ˜ ε | ≤ C ( L, δ ) unless θ ′ L ( h δ,ε ( z ˜ ε )) = 0.Using the local ǫ -H¨older regularity of R i , we then compute for | z ˜ ε | ≤ C ( L, δ ) D i h ( z ˜ ε ) R i ( z ) = D i h ( z ˜ ε )[ R i ( z ) − R i ( z ˜ ε )] + D i h ( z ˜ ε ) R i ( z ˜ ε ) ≤ C ( L, δ )(1 + n X i =1 log( c i + ˜ ε ))˜ ε ǫ + D i h ( z ˜ ε ) R i ( z ˜ ε ) . The first term in the last line converges, as ˜ ε →
0, uniformly to zero on the set {| z ˜ ε | ≤ C ( L, δ ) } . The second term is non-positive by hp. (A2.i). Thus, since θ ′ L ≥ L →∞ θ ′ L ( s ) = 1 for all s ∈ R we can use Fatou’s lemma to inferlim sup L →∞ lim sup ˜ ε → ˆ T ˆ Ω θ ′ L ( h δ,ε ( z ˜ ε )) D i h ( z ˜ ε ) R i ( z ) d x d t ≤ lim sup L →∞ ˆ T ˆ Ω θ ′ L ( h δ,ε ( z )) D i h ( z ) R i ( z ) d x d t ≤ ˆ T ˆ Ω D i h ( z ) R i ( z ) d x d t. Observe that the last line is independent of δ and ε .Put together, the above inequalities and equation (4.4) imply the entropy dissipationinequality ( ed ).Case 2: 0 < s < t < T ∗ .We assert that the fact that z is a renormalised solution in Ω T ∗ in the sense of Defini-tion 2.1 implies that for a.e. 0 < s < t < T ∗ , and all ˜ ψ ∈ C ∞ ([0 , T ∗ ) × ¯Ω) ˆ Ω ξ ( z ( t, · )) ˜ ψ ( t, · ) d x − ˆ Ω ξ ( z ( s, · )) ˜ ψ ( s, · ) d x − ˆ ts ˆ Ω ξ ( z ) ∂ τ ˜ ψ d x d τ = − ˆ ts ˆ Ω D ij ξ ( z ) A ik ( z ) ∇ z k · ∇ z j ˜ ψ d x d τ (4.8) − ˆ ts ˆ Ω D i ξ ( z ) A ik ( z ) ∇ z k · ∇ ˜ ψ d x d τ + ˆ ts ˆ Ω D i ξ ( z ) R i ( z ) ˜ ψ d x d τ for all ξ ∈ C ∞ ( R n ≥ ). EAK-STRONG UNIQUENESS FOR ERDS 35
This can be proved as follows: take η ∈ C ∞ ( R ), η ′ ≥ η = 0 on ( −∞ , − η = 1on [0 , ∞ ), and define η s,δ ( τ ) := η ( τ − sδ ) for 0 < δ ≪
1. Observe that η s,δ ( τ ) = 0 for τ ≤ s − δ and η s,δ ( τ ) = 1 for τ ≥ s . In the renormalised formulation (2.1) with T = t ,we choose the test function ψ ( τ, x ) := ˜ ψ ( τ, x ) η s,δ ( τ ). Then, the corresponding right-handside of (2.1) converges, as δ →
0, to the right-hand side of eq. (4.8) by the dominatedconvergence theorem. The corresponding left-hand side takes the form ˆ Ω ξ ( z ( t, · )) ˜ ψ ( t, · ) d x − ˆ ss − δ ˆ Ω ξ ( z ) ˜ ψ d x ∂ τ η s,δ d τ + O ( δ ) − ˆ ts ˆ Ω ξ ( z ) ∂ τ ˜ ψ d x d τ. The second term in the last line can be rewritten as (cid:0) F ∗ ( δ η ′ ( − · δ )) (cid:1) ( s ) for the measur-able, bounded function F ( τ ) := − ´ Ω ξ ( z ( τ, x )) ˜ ψ ( τ, x ) d x , and since ´ R η ′ ( − τ )d τ = 1,we have the convergence (cid:0) F ∗ ( δ η ′ ( − · δ )) (cid:1) ( s ) → F ( s ) = − ´ Ω ξ ( z ( s, x )) ˜ ψ ( s, x ) d x fora.e. s ∈ (0 , T ∗ ) as δ →
0. This establishes the asserted identity (4.8).From (4.8) we infer that, for a.e. 0 < s < T ∗ , the function ( τ, x ) z ( s + τ, x ) is arenormalised solution in (0 , T ∗ − s ) × Ω with initial data z ( s, · ). Now we can invoke Case 1and deduce ( ed .s).5. Exponential convergence to equilibrium
Proof of Proposition 2.10.
Let us first observe that the regularity hypotheses on z =( u, c ) together with the bounds ( ed .s) and ( ene .s) imply that P ( z ) + |∇ u | ∈ L (Ω T )for any T < ∞ and that u ∈ L ∞ loc ([0 , ∞ ) , L (Ω)), c i log c i ∈ L ∞ loc ([0 , ∞ ) , L (Ω)) for all i ∈ { , . . . , n } . Here, we also used the lower and upper bounds on H ( z ) provided inLemma 6.3.The energy and mass conservation properties ´ Ω z l ( t, x ) d x = ¯ z l for a.e. t >
0, where l ∈ { , , . . . , n } , can be seen as follows. In the renormalised formulation (2.1), we choose ψ ≡ ξ ( z ) := ϕ El ( z ) for E ≥
1, where ϕ El ( z ) = Eϕ l ( E − z ) for some ϕ l ∈ C ∞ ( R n ≥ )with supp Dϕ l compact and ϕ l ( z ) = z l for | z | < ϕ l ).This gives ˆ Ω ϕ El ( z ( T, · )) d x − ˆ Ω ϕ El ( z in ) d x = − ˆ T ˆ Ω D ij ϕ El ( z ) A ik ( z ) ∇ z k · ∇ z j d x d t. (5.1)By the dominated convergence theorem, the LHS converges, as E → ∞ , to ˆ Ω z l ( T, x ) d x − ˆ Ω z in l d x. Since D ϕ l ( z ) = 0 for | z | <
1, we have lim E →∞ D ij ϕ El ( z ) = 0 for every z ∈ R n ≥ . At thesame time, using the bounds in Lemma 6.1 it is easy to see that for all i, j ∈ { , . . . , n }| D ij ϕ El ( z ) n X k =0 A ik ( z ) ∇ z k · ∇ z j | . P ( z ) + |∇ u | uniformly in E ≥
1. Hence, the integral on the RHS of (5.1) converges to zero as E → ∞ thanks to dominated convergence.Let us now sketch the proof showing the exponential convergence to equilibrium. Below, ǫ k , k = 0 , , . . . , denote fixed positive constants. As we only consider Model (M0) , it suffices to take α ∈ (0 , z being spatially constant, yields for a.e. time H rel ( z, ¯ z ) = H ( z ) − D i h (¯ z ) ˆ Ω ( z i − ¯ z i ) d x − H (¯ z )= H ( z ) − H (¯ z ) . Hence, inequality ( ed .s) gives for a.e. t ≥ s ≥ H rel ( z ( τ ) , ¯ z ) (cid:12)(cid:12)(cid:12)(cid:12) τ = tτ = s ≤ − ˆ ts ˆ Ω P ( z ) d x d τ. Writing B ( c, ¯ c ) := P ni =1 b ( c i , ¯ c i ), where b ( s, ¯ s ) = ¯ sλ ( s/ ¯ s ), we have by estimate (6.2) andthe logarithmic Sobolev inequality (cf. [39]) ˆ Ω P ( z ) d x ≥ ǫ n X i =1 ˆ Ω |∇√ c i | d x ≥ ǫ ˆ Ω B ( c, ¯ c ) d x, (5.2)where ǫ = ǫ (Ω) > g rel in (3.4), and using ´ u d x = ´ ¯ u d x , ( ene .s), and thePoincar´e–Wirtinger inequality, we can further estimate for some ǫ = ǫ (Ω) > G rel ( u ( τ ) , ¯ u ) (cid:12)(cid:12)(cid:12)(cid:12) τ = tτ = s ≤ − ǫ ˆ ts ˆ Ω | u − ¯ u | d x d τ + ̟C ˆ ts ˆ Ω P ( z ) d x d τ. (5.3)We next let E ≥ | ¯ z | , to be fixed later. Then, by uniform convexity, for | z | ≤ E , B ( c, ¯ c ) + α | u − ¯ u | & E,α | z − ¯ z | & E h rel ( z, ¯ z ) . (The argument leading to the second inequality is as in the proof of (3.5).)At the same time, B ( c, ¯ c ) + α | u − ¯ u | ≥ n X i =1 c i log c i + αu ! − C (¯ z ) . Hence, for E = E (¯ z, α ) large enough, we obtain B ( c, ¯ c ) + α | u − ¯ u | & E h rel ( z, ¯ z ) . Combining the above estimates and choosing α ∈ (0 ,
1] such that αC̟ ≤ (with C asin (5.3)), we infer (cid:20) H rel ( z ( τ ) , ¯ z ) + αG rel ( u ( τ ) , ¯ u ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) τ = tτ = s ≤ − ˆ ts ˆ Ω P ( z ) d x d τ − ǫ α ˆ ts k u − ¯ u k L (Ω) d τ ≤ − ǫ ˆ ts (cid:20) H rel ( z ( τ ) , ¯ z ) + αG rel ( u ( τ ) , ¯ u ) (cid:21) d τ, where ǫ = ǫ (¯ z, α, Ω) >
0. A version of Gronwall’s inequality (see e.g. [21, p. 702]) yieldsthe asserted bound (2.13) for λ = ǫ > π γ |∇ u | is dom-inated by P ( z ) (see (5.2)). If σ ( u ) is sufficiently close to a linear function for u ≫ σ ( u ) = u − ǫ ( d ) for ǫ ( d ) > EAK-STRONG UNIQUENESS FOR ERDS 37 Appendix
Lemma 6.1 (Estimates for Model (M0) ) . Let the hypotheses of Model (M0) be satisfied.Then, formally, n X k =0 A k ( z ) ∇ z k = a ( z ) ∇ z + m ( z ) ∇ D h ( z ) , (6.1a) n X k =0 A ik ( z ) ∇ z k = m i ( z ) ∇ D i h ( z ) + a ( z ) c i w ′ i ( u ) w i ( u ) ∇ u for i ≥ , (6.1b) where a ( z ) = π ( z ) γ ( z ) and γ is given by (2.6) .Moreover, for any sufficiently regular function z = ( u, c , . . . , c n ) with positive compon-ents, we have the following estimates:Abbreviating P ( z ) := ∇ z : ( D h ( z ) A ( z ) ∇ z ) = ∇ D i h ( z ) · ( M il ( z ) ∇ D l h ( z )) and γ ( u, c ) := − ˆ σ ′′ ( u ) − P nl =1 w ′′ l ( u ) w l ( u ) c l , one has P ( z ) & n X i =1 |∇√ c i | + |√ γ ∇ u | + |√ m ∇ D h ( z ) | , (6.2) and a ( z ) |∇ u | ∼ |∇ u | . (6.3) Furthermore, | A ( z ) ∇ z | . (cid:16) max i =1 ,...,n √ c i + p π ( z ) + p m ( z ) (cid:17)p P ( z ) , (6.4) | n X k =0 A ik ( z ) ∇ z k | . √ c i p P ( z ) for i ≥ , (6.5) | n X k =0 A k ( z ) ∇ z k | . |∇ u | + √ m p P ( z ) . (6.6) Proof.
Identities (6.1a)–(6.1b) follow from a straightforward computation using the defin-ition of µ i (see [24], if necessary). Except for the last term in (6.2), estimate (6.2) is a con-sequence of [24, Lemma 2.1] (in [24]: m ≡ |√ m ∇ D h ( z ) | for m = m ( z ) ≥ P ( z ) and M . Eq. (6.3) is im-mediate since a = πγ ∼ m ≡
0, and the current version thus follows estimate (6.2),which implies the bound | m ( z ) ∇ D h ( z ) | ≤ √ m p P ( z ) . Estimate (6.5) is a consequenceof the proof of [24, Lemma 2.3], while estimate (6.6) follows from the fact that a ∼ m ( z ) ∇ D h ( z ) observed before. Lemma 6.2.
Model (M0) (see page 13) fulfils conditions (A1) – (A5) and (A6) of The-orem 2.8 when assuming additionally the regularity hypotheses ˆ σ, w i ∈ C ((0 , ∞ )) , (A2.ii) and m, m i , π ∈ C , ((0 , ∞ ) n ) .Proof. The asserted estimates can be verified using Lemma 6.1: the bounds in (A4) areimmediate consequences of estimates (6.2), (6.4) combined with the bound 0 ≤ p π ( z ) . (1 + u ). Condition (A5) follows from estimating | a ( z ) ∇ u | . √ π |√ γ ∇ u | and using (6.2).Condition (A6) easily follows from (6.5) and (6.6). (We have not aimed at optimisingthe conditions on m ( z ), which are far from being sharp.) The conditions in (A3) followfrom the definition of M in (2.7). Lemma 6.3 (Lower and upper entropy bounds) . Let h = h ( u, c ) be given by (h1) with (h2) being satisfied. There exist positive constants ǫ β > , κ β ∈ (0 , and C β , C ∈ (0 , ∞ ) such that for all ( u, c ) ∈ [0 , ∞ ) n h ( u, c ) ≥ − ˆ σ ( u ) + ǫ β n X i =1 c i log( c i ) − Cu κ β − C β , (6.7) h ( u, c ) ≤ − ˆ σ ( u ) + C n X i =1 c i log( c i ) + C. (6.8) Proof.
Letting β ∗ = β +12 ∈ ( β, c i log( w i ( u )) = c i log( w i ( u )) χ { w i ( u ) ≤ c β ∗ i } + c i log( w i ( u )) χ { w i ( u ) >c β ∗ i } ≤ β ∗ c i log( c i ) + w i ( u ) β ∗ log( w i ( u )) + C ≤ β ∗ c i log( c i ) + Cu (1+ β/β ∗ ) + C. Thus, h ( u, c ) = − ˆ σ ( u ) + n X i =1 (cid:0) λ ( c i ) − c i log( w i ( u )) (cid:1) ≥ − ˆ σ ( u ) + n X i =1 (cid:0) (1 − β ∗ ) c i log( c i ) − c i + 1 (cid:1) − Cu (1+ β/β ∗ ) − C. This yields (6.7) with κ β := (1 + β/β ∗ ) < ǫ β = (1 − β ∗ ) > C β < ∞ .Estimate (6.8) easily follows from the hypothesis that w i (0) > i (see also theproof of [24, eq. (2.9)]). Lemma 6.4 (Minimum principle) . In addition to the hypotheses of Theorem 2.8 assumethat ̟ = 0 . Let T ∈ (0 , T ∗ ) and u := inf Ω T u in > . Then the energy component u of therenormalised solution z = ( u, c ) satisfies u ≥ u almost everywhere in Ω T .Sketch proof. The hypotheses imply that u ∈ L ∞ loc ( I ; L (Ω)) and that there exists r > a ( z ) ∇ u ∈ L r loc ( I ; L r (Ω)), ∂ t u ∈ L r loc ( I ; ( W ,r ′ (Ω)) ∗ ), r + r ′ = 1. The weakformulation of the energy component (2.3) can therefore be integrated by parts withrespect to time to give ˆ T ′ h ∂ t u, ϕ i d t = − ˆ T ′ ˆ Ω a ( z ) ∇ u · ∇ ϕ d x d t. (6.9)for a.a. T ′ ∈ (0 , T ]. Ignoring regularity issues for the moment and testing the equationwith ϕ = ( u − u ) − leads to12 ˆ Ω | ( u − u ) − | d x (cid:12)(cid:12)(cid:12)(cid:12) t = T ′ t =0 + ˆ T ′ ˆ Ω a ( z ) |∇ ( u − u ) − | d x d t = 0 . (6.10)This implies that ( u − u ) − = 0 and hence u ≥ u a.e. in Ω T .To make the argument rigorous, one considers a smooth partition of unity ( χ k ) Nk =1 on¯Ω as in the proof of the L identities in [24, Lemma 6.1], see also [11, Lemma 12] and [22,Lemma 4]. For simplicity, we only outline the reasoning in the case of ψ := χ k beingcompactly supported in Ω, and refer, for the general case, to the first and the third ofthe references provided before. EAK-STRONG UNIQUENESS FOR ERDS 39
Denote by ˜ ρ the standard mollifying kernel, let ρ := ˜ ρ ∗ ˜ ρ and ρ ε ( x ) := ε d ρ ( xε ). Then,for ε > ψ, ∂ Ω)), choose in (6.9) the testfunction ϕ = ρ ε ∗ (( ρ ε ∗ u − u ) − ψ ) , which lies in W ,r loc ( I ; H s (Ω)) for any s ∈ N . Abbreviate u ε = ρ ε ∗ u and compute ∂ t u ε ( u ε − u ) − ψ =
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