On evolutionary problems with a-priori bounded gradients
aa r X i v : . [ m a t h . A P ] F e b ON EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS
MIROSLAV BUL´IˇCEK, DAVID HRUˇSKA, AND JOSEF M ´ALEK
Abstract.
We study a nonlinear evolutionary partial differential equation that can be viewed as a generalizationof the heat equation where the temperature gradient is a priori bounded but the heat flux provides merely L -coercivity. Applying higher differentiability techniques in space and time, choosing a special weighted norm(equivalent to the Euclidean norm in R d ), incorporating finer properties of integrable functions and using theconcept of renormalized solution, we prove long-time and large-data existence and uniqueness of weak solution,with an L -integrable flux, to an initial spatially-periodic problem for all values of a positive model parameter.If this parameter is smaller than 2 / ( d + 1), where d denotes the spatial dimension, we obtain higher integrabilityof the flux. As the developed approach is not restricted to a scalar equation, we also present an analogous resultfor nonlinear parabolic systems in which the nonlinearity, being the gradient of a strictly convex function, givesan a-priori L ∞ -bound on the gradient of the unknown solution. Introduction
Problem setting and main result.
This paper concerns a parabolic-like problem involving nonlinearelliptic operators that can be viewed as regularizations of the ∞ -Laplacian. More precisely, for fixed L >
T > , L ) d ⊂ R d and Q := (0 , T ) × Ω and investigate the following problem: for givenΩ-periodic functions g : [0 , T ] × R d → R , u : R d → R and a given parameter a >
0, find an Ω-periodic function u : [0 , T ] × R d → R and a vectorial Ω-periodic function q : [0 , T ] × R d → R d such that ∂ t u − div q = g in Q, (1.1a) ∇ u = q (1 + | q | a ) a in Q, (1.1b) u (0 , · ) = u in Ω . (1.1c)The motivation for investigating such type of problems is given below. The main result of this paper is thefollowing: for sufficiently smooth initial data u , which satisfies a reasonable compatibility condition, and forsufficiently smooth right-hand side g , there exists a unique couple ( u, q ) solving (1.1) in the sense of distributions. To formulate the result precisely, we need to fix the notation, the appropriate function spaces and the conceptof solution to (1.1). Since we are dealing with a spatially periodic problem, we recall the definition of periodicSobolev spaces W k,pper (Ω) := (cid:26) u = ˜ u (cid:12)(cid:12) Ω , ˜ u ∈ C ∞ ( R d ) is Ω-periodic (cid:27) k·k k,p , where k ∈ N and p ∈ [1 , ∞ ) are arbitrary (note that L per (Ω) = L (Ω) and that these spaces, as closed subspacesof reflexive Banach spaces, are reflexive as well provided that p ∈ (1 , ∞ )). The space W k, ∞ per is then defined as W k, ∞ per (Ω) := W k, per (Ω) ∩ W k, ∞ (Ω) . Throughout the paper, we use standard notation for Lebesgue, Sobolev and Bochner spaces equipped with theusual norms. Unless stated otherwise, bold letters, e.g. q , are used for vector-valued functions to distinguishthem from scalar functions. The symbol “ ∂ t ” stands for the partial derivative with respect to the time variable t ∈ (0 , T ), while the operators “ ∇ ” and “div” take into account only the spatial variables ( x , . . . , x d ) ∈ Ω.Later, we also use “ ∂ j ” to abbreviate partial derivative with respect to x j . The shortcut “a.e.” abbreviates almost everywhere and “a.a.” stands for almost all .Next, we define the notion of a weak solution to (1.1) and formulate the main result. Mathematics Subject Classification.
Key words and phrases. nonlinear parabolic equation, weak solution, existence, uniqueness, renormalized solution, ∞ -Laplacian,a priori bounded gradient.M. Bul´ıˇcek’s work is supported by the project 20-11027X financed by Czech science foundation (GAˇCR). J. M´alek acknowledgesthe support of the project No. 18-12719S financed by Czech Science Foundation (GAˇCR). M. Bul´ıˇcek and J. M´alek are members ofthe Neˇcas Center for Mathematical Modeling. The PhD position of D. Hruˇska is funded by the German Science Foundation DFGin context of the Priority Program SPP 2026 “Geometry at Infinity”. Definition 1.1.
Let u ∈ L (Ω) , g ∈ L ( Q ) and a > . We say that a couple ( u, q ) is a weak solution toproblem (1.1) if u ∈ W , (cid:0) , T ; L (Ω) (cid:1) ∩ L (cid:0) , T ; W , per (Ω) (cid:1) , q ∈ L (cid:0) , T ; L (cid:0) Ω; R d (cid:1)(cid:1) and Z Ω ∂ t u ϕ + q · ∇ ϕ d x = Z Ω g ϕ d x for all ϕ ∈ W , ∞ per (Ω) and a.a. t ∈ (0 , T ) , (1.2a) ∇ u = q (1 + | q | a ) a a.e. in Q, (1.2b) k u ( t, · ) − u k L (Ω) t → + −−−−→ . (1.2c) Theorem 1.2.
Let a > , g ∈ L (cid:0) , T ; L (Ω) (cid:1) and u ∈ W , ∞ per (Ω) satisfy (1.3) k∇ u k L ∞ (Ω) =: U < . (i) Then there exists a unique weak solution to problem (1.1) in the sense of Definition 1.1. Moreover, thesolution satisfies (1.4) u ∈ L (cid:0) , T ; W , per (Ω) (cid:1) . (ii) Furthermore, if g ∈ W , (cid:0) , T ; L (Ω) (cid:1) and u ∈ W , per (Ω) , then u ∈ W , ∞ (0 , T ; L (Ω)) . If, in addition, theparameter a satisfies (1.5) a ∈ (cid:18) , d + 1 (cid:19) , then (1.6) q ∈ L b ( Q ; R d ) for ( b = (1 − a )( d +1) d − > if d ≥ ,b arbitrary if d = 1 . The paper is structured in the following way. In the rest of this section, we describe the main novelties of ourresult in detail. We also add a physical motivation for studying such problems and show the key difficulties of thestudied problem. Section 2 contains several auxiliary results needed in the proof of Theorem 1.2. In Section 3,we prove the uniqueness result. Sections 4 and 5 concern the existence result. In Section 4, we introduce asuitable ε -approximation of the problem (1.1), which is then treated by the standard Faedo-Galerkin method incombination with a cascade of energy estimates that helps to establish the existence of a weak solution to the ε -approximation for arbitrary fixed ε ∈ (0 , ε . Then, in Section 5, letting ε → renormalizationtechnique together with a special choice of weigthed scalar product (equivalent to the standard scalar product in R d ) to identify a weak solution of the original problem. Section 6.2 is devoted to the proof of higher regularity(integrability) of the flux q for the values of a satisfying (1.5), which concludes the proof of the second part ofTheorem 1.2. In the final section, we formulate a generalization of the results stated in Theorem 1.2.1.2. State of the art and main novelties.
In order to put our result in an appropriate context, we intro-duce nonlinear (quasilinear) elliptic and parabolic problems characterized by the presence of p -Laplacian or itsgeneralizations of various forms. Thus, for d ∈ N , a > δ ∈ { , } and p satisfying 1 < p ≤ ∞ , we define f p ′ : R d → R d by(1.7) f p ′ ( q ) := ( δ + | q | a ) p ′− a q , where p ′ = ( pp − if p ∈ (1 , ∞ ) , p = ∞ . Similarly, now for p satisfying 1 ≤ p < ∞ , we set g p : R d → R d as(1.8) g p ( z ) := ( δ + | z | a ) p − a z . Replacing the equation (1.1b) by(1.9) ∇ u = f p ′ ( q ) with f p ′ introduced in (1.7) , we obtain(1.10) ∂ t u − div q = g in Q, ∇ u = ( δ + | q | a ) p ′− a q in Q,u (0 , · ) = u in Ω , N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 3 while replacing (1.1b) by(1.11) q = g p ( ∇ u ) with g p introduced in (1.8) , we end up with(1.12) ∂ t u − div (cid:16) ( δ + |∇ u | a ) p − a ∇ u (cid:17) = g in Q,u (0 , · ) = u in Ω . Next, let us first restrict ourselves to the case p ∈ (1 , ∞ ). Then, the mappings f p ′ and g p are strictly monotonefor all a > δ ∈ { , } . In addition, when δ = 0, f p ′ = ( g p ) − and (1.10) and (1.12) coincide. Note thatwhen δ = 1 the ( q , ∇ u )-relations are smoothed out near zero (thus eliminating the degeneracy/singularity of thecorresponding elliptic operator) and the problems (1.10) and (1.12) do not describe the same ( q , ∇ u )-relationanymore. In all these cases the natural function spaces for the solution are as follows: u ∈ L p (cid:0) , T ; W ,pper (Ω) (cid:1) ∩ W ,p ′ (cid:0) , T ; W ,pper (Ω) ∗ (cid:1) , q ∈ L p ′ (cid:16) , T ; L p ′ (cid:0) Ω; R d (cid:1)(cid:17) , provided that the data satisfy u ∈ L per (Ω) and g ∈ L p ′ (0 , T ; W ,pper (Ω) ∗ ). Within this functional setting, theexistence and uniqueness theory for such problems is nowadays classical, see [20, 23] including and extendingthe monotone operator theory invented by Minty for the elliptic setting in Hilbert spaces (see [25]). It turns outthat one can develop a rather complete theory for such problems and we refer to the classical monograph [15] foradditional regularity results. Furthermore, one can introduce a much more general class of possible relationshipsbetween q and ∇ u that goes far beyond (1.9) or (1.11) and where q and ∇ u are related implicitly. This meansthat instead of (1.1b) one considers the equation g ( q , ∇ u ) = in Q with g : R d × R d → R d continuous. Undersuitable assumptions imposed on g , providing among others p -coercivity for ∇ u and p ′ -coercivity for q , a self-contained large-data mathematical theory within the above functional setting has been recently developed, alsofor the systems, in [13] (including, but also extending the results established in [11, 12] in the context of fluidmechanics).A natural and interesting question is what happens when p → + or p → ∞ . In the case δ = 0, we formallyobtain from (1.11) for p = 1 that q = ∇ u |∇ u | . Then, the governing equation for the time-independent (stationary) problem being of the form − div( ∇ u/ |∇ u | ) = g formally represents the Euler-Lagrange equation corresponding to the minimization of the total variation func-tional. Analogously, and again for δ = 0, it follows from (1.9) that for p = ∞ (i.e. p ′ = 1) one has ∇ u = q | q | , which, together with the governing equation − div q = g , corresponds to the so-called ∞ -Laplacian, see alsoFig. 1. |∇ u || q | p = 11 1 p = p = 2 p = 3 p = + ∞ Figure 1. If p ∈ (1 , ∞ ), then q = |∇ u | p − ∇ u ⇔ ∇ u = | q | p ′ − q with p ′ = pp − . Selectedgraphs are drawn (for values p = , , p = 1 and p = ∞ (i.e. p ′ = 1) aresketched as well. M. BUL´IˇCEK, D. HRUˇSKA, AND J. M´ALEK
Both limiting cases have attracted attention in the scientific community. Not only is the understanding ofthese limiting cases interesting as a mathematical problem per se , but also the total variation equation or ∞ -Laplacian are frequently used when studying sharp interface-like problems, image recovering, etc. Let us pointout that, in the elliptic (i.e. stationary) setting, one faces serious difficulties with defining a proper concept ofsolution and usually one has to introduce a new one. While for p = 1 this has led to the theory of BV spaces,see e.g. [19], for p = ∞ the concept of viscosity solution was introduced in [3]. In principle, one can say that theexpected L -regularity for ∇ u (when p = 1) or the L -regularity for q (when p = ∞ ) must be relaxed and oneis led to work in the “weak ∗ closure of L ” or, more precisely, in the space of Radon measures. In the parabolicsetting, there is a certain mollification effect coming from the presence of the time derivative and therefore thecase p = 1 is not so difficult to treat provided that the initial data are sufficiently regular, see e.g. [2]. However,for p = ∞ , one seems to be forced to keep the notion of a viscosity solution, see [1, 26]. Furthermore, it is alsowell known that the viscosity solution is in principle the best object one can deal with, which is well documentedby the existence of a singular solution (see [4] or the monograph [22]).The above discussion was focused on the case δ = 0, which leads to certain singular behaviour near zero. Fora mollified problem with δ = 1, the limiting cases take the form q = ∇ u (1 + |∇ u | a ) a for p = 1 , ∇ u = q (1 + | q | a ) a for p = ∞ , which may have better properties since both equations represent strictly monotone mapping unlike the case δ = 0, see also Fig. 2. Nevertheless, even in this regularized case, one encounters difficulties. The most famousexample concerns the case a = 2 and p = 1, i.e. the minimal surface problem. Due to Finn’s counterexample(see [16]), it is known that even for smooth data one can obtain an irregular solution that is not a Sobolevfunction. However, such a singularity appears only on (the Dirichlet part of) the boundary. This follows fromtwo results: the interior regularity established for the stationary problem with p = 1 and a ≤ p = 1 and a > p = ∞ and δ = 1. |∇ u || q | p = 1 p = p = 2 p = 3 p = 10 |∇ u || q | p ′ = 1 p ′ = p ′ = 2 p ′ = 3 p ′ = 10 Figure 2.
On the left, the graphs of q = (1 + |∇ u | ) p − ∇ u are sketched for selected valuesof p ∈ [1 , ∞ ), namely p = 1 , , , ,
10. On the right, the graphs of ∇ u = (1 + | q | ) p ′− q areshown for p ′ = 1 , , , , ∞ -Laplacian and try to treat the problem with thenotion of viscosity solution. However, it is not clear how to adopt the theory of viscosity solution to our settingsince we are dealing with a different elliptic operator (compare the limiting behaviour for p = ∞ and δ = 0 or δ = 1 depicted at Figures 1 and 2). More importantly, it turns out (and this is one of the main messages ofthis paper) that we do not need to introduce the concept of viscosity solution as we are able to establish theexistence of a standard weak solution. Our method builds on the approach developed in [10] and [7], wherea similar elliptic problem arising in solid mechanics is analyzed. In this paper, we generalize the approachproposed in [10, 7] (and used in some sense also in [6]) and adopt it to the parabolic setting.An interesting problem might be the study of the limit a → ∞ . In such a case(1 + | q | a ) a ց max { , | q |} as a → ∞ and consequently (for f introduced in (1.7)) f ( q ) = q (1 + | q | a ) a ր q | q | min { , | q |} as a → ∞ . N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 5
However, the limiting mapping is not strictly monotone (see Fig. 3) and the method developed in this papercannot be applied. | q ||∇ u | a = a = 1 a = 2 a = 6 a = + ∞ Figure 3.
The graphs of ∇ u = q (1+ | q | a ) a are drawn for selected values of parameter a ∈ (0 , ∞ ).The limiting case a = ∞ is sketched as well.To summarize and emphasize the novelty of our result once again, we show the existence of a weak solutionto the evolutionary problem (1.1) for all a > q being an integrable function.It is worth mentioning that our proof of Theorem 1.2, as presented below, is based on two properties of thenonlinear function f defined in (1.7), namely, its radial structure, i.e. f ( q ) = α ( | q | ) q , and the existence ofstrictly convex potential to f . Consequently, the specific form of the equation (1.1b) is not essential and wecan develop a satisfactory theory for a general class of relations behaving like mollified ∞ -Laplacian (providedthat there is a strictly convex potential behind). We state such a generalized result in Theorem 7.1 in Section 7but do not provide the proof for simplicity here. However, an interested reader can compare our proof with thegeneral methods invented in [8] for the elliptic setting. In fact, by adopting these methods and combining themwith the proof of Theorem 1.2, one can prove Theorem 7.1.1.3. A fluid mechanics problem motivating this study.
Consider an incompressible fluid with constantdensity flowing, at a uniform temperature, in a three-dimensional domain. In the absence of external bodyforces, unsteady flows of such a fluid are described by the following set of equations for the unknown velocityfield v = ( v , v , v ) and the pressure p :(1.13) div v = 0 , ∂ t v + X k =1 v k ∂ k v = −∇ p + div S , where S , the deviatoric part of the Cauchy stress tensor, enters the additional (so-called constitutive) equationrelating S to the symmetric part of the velocity gradient denoted by D and characterizing the material propertiesof a particular class of fluids. While for the Newtonian fluids one has S = 2 ν ∗ D , where ν ∗ > S and D is nonlinear.There are fluids (see for example [21, 29, 18, 17, 27]) in which the constitutive relation capable of describingexperimental data can be of the form(1.14) 2 ν ∗ D = S (cid:16) (cid:16) √ | S | (cid:17) a (cid:17) a for some a > ν ∗ > . The general goal is to understand mathematical properties associated with the system of partial differentialequations (1.13)-(1.14). A possible natural approach is to look first at a geometrically simplified version of theproblem. For example, one can investigate simple shear flows taking place between two infinite parallel plateslocated at x = 0 and x = L . Time-dependent simple shear flows are characterized by the velocity field of theform v ( t, x , x , x ) = ( u ( t, x ) , , v = 0. We also infer that the onlynontrivial components of D are D = D = ∂ u . Hence it follows from (1.14) that also all components of S other than S = S =: σ = σ ( t, x ) vanish. Then the second equation in (1.13) together with (1.14) leads to: ∂ t u = − ∂ p + ∂ σ, − ∂ p, − ∂ p, (1.15a) ν ∗ ∂ u = σ (1 + | σ | a ) a . (1.15b)It follows from the second and the third equation in (1.15a) that p = p ( t, x ). After inserting this piece ofinformation into the first equation of (1.15a) we can decompose this equation and obtain(1.16) ( ∂ t u − ∂ σ )( t, x ) = g ( t ) and − ∂ p ( t, x ) = g ( t ) M. BUL´IˇCEK, D. HRUˇSKA, AND J. M´ALEK for some function g depending only on time. When studying the unsteady Poiseuille flow, the function g ,corresponding to the pressure drop, must be given. Then the first equation in (1.16) together with (1.15b)represents a one-dimensional version of the governing equations of the problem (1.1) studied in this paper (withthe caveat that in (1.1) the function g may also depend on the spatial variable).1.4. Difficulties and main idea.
As mentioned above, the key difficulty is due to a weak a priori estimatefor q compensating the fact that ∇ u is bounded a priori. To be more explicit, let us recall the definition (1.7)with δ = 1, i.e. f ( q ) := q (1+ | q | a ) a . Obviously, | f ( q ) | = | q | (1+ | q | a ) a < q ∈ R d . This directly yieldsthat ∇ u ∈ L ∞ ( Q ; R d ), but it also brings the restriction that the inverse function of (the injective function) f cannot be defined outside of the unit ball in R d and hence we may not simply write q as a function of ∇ u anddirectly apply the Faedo–Galerkin approximation method.Next, standard energy estimates are not sufficient to establish the existence of a weak solution. Indeed,multiplying the linear equation (1.1a) by the solution u , integrating by parts with respect to the spatial variables(the spatial periodicity ensures that the boundary terms vanish) and substituting for ∇ u from (1.1b) we concludethat Z Q | q | (1 + | q | a ) a d x d t < ∞ . However, this implies merely that q belongs to L ( Q ; R d ) which is not a reflexive Banach space (it does noteven have a predual). Hence, when constructing a solution, we may not identify a weak limit of a subsequenceof { q n } ∞ n =1 , a sequence of some approximations bounded in L ( Q ; R d ). Similar difficulties occur if one aims toinvestigate the limiting behaviour when converging from the p -Laplacian to the ∞ -Laplacian, i.e. when studyingthe limit p ′ →
1+ in (1.7).At this point one might consider a priori estimates involving higher derivatives. Let us denote by s a generaltime or spatial variable, i.e. s can represent t, x , . . . , x d . Let us differentiate the equation (1.1a) with respectto s , multiply the result by ∂ s u and integrate over Ω. Finally, in the integral involving q , we integrate by partsand obtain 12 dd t k ∂ s u k L (Ω) + Z Ω ∂ s q · ∂ s ( ∇ u ) d x = Z Ω ∂ s g∂ s u d x. Hence, if the data are sufficiently regular, one can hope for an a priori estimate for q of the form(1.17) Z Q ∂ s q · ∂ s ( ∇ u ) d x d t < ∞ . Let us now focus on the information coming from (1.17) for general f p ′ with p ′ ∈ [1 , ∞ ). Using (1.7) (cf.Lemma 2.1) one obtains(1.18) ∂ s q · ∂ s ( ∇ u ) = (1 + | q | a ) p ′− − aa (cid:16) | ∂ s q | (1 + | q | a ) + ( p ′ − | q | a − ( q · ∂ s q ) (cid:17) . For p ′ > p ′ − > − ∂ s q · ∂ s ( ∇ u ) ≥ C (1 + | q | a ) p ′− a | ∂ s q | , where C := min { p ′ − , } > ∂ s q in L s ( Q ; R d ) for some s >
1. However,in the critical case p ′ = 1, there is a sudden loss of information as one then deduces merely the estimate(1.19) ∂ s q · ∂ s ( ∇ u ) ≥ (1 + | q | a ) − − aa | ∂ s q | . Consequently, the power of | q | in this weighted estimate drops by a . For small values of a , namely for thosesatisfying (1.5), it can be deduced from (1.17) and (1.19) using Sobolev embedding that q is bounded in L b ( Q ; R d ) for some b >
1, see (1.6). This is shown in the proof of the second part of Theorem 1.2. However,for large values of a , the estimate (1.19) seems to be useless at the first glance. We will however show that itimplies almost everywhere convergence for a selected subsequence of { q m } . This is still not sufficient to takethe limit in the governing equation (due to L -integrability of { q m } ). This is why we introduce the conceptof renormalized solution for a suitable m -approximating problem and then, in order to take the limit from therenormalized formulation of the approximate problem to the weak formulation of the original problem, we shallwork directly with the quantity ∂ s q · ∂ s ( ∇ u ) (or more precisely with the right-hand side of (1.18)), which insome sense still generates an estimate for ∂ s q in some scalar product in R d induced by q itself. N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 7 Preliminaries
Here and in the remaining parts of this text we set, for a > f ( q ) := q (1 + | q | a ) a where q ∈ R d . The aim of this section is to collect basic properties of f as well as its ε -approximation f ε defined, for ε > f ε ( q ) := f ( q ) + ε q = q (1 + | q | a ) a + ε q . Lemma 2.1.
The following assertions hold true:(i) f , f ε ∈ C ( R d ; R d ) and for all i, j = 1 , . . . , d and arbitrary q ∈ R d there holds: (2.3) ( ∇ q f ( q )) ij := ∂f i ( q ) ∂q j = (1 + | q | a ) δ ij − | q | a − q i q j (1 + | q | a ) a and ( ∇ q f ε ( q )) ij = ( ∇ q f ( q )) ij + εδ ij , where δ ij is the Kronecker delta.(ii) Introducing the scalar functions f ( s ) := s (1+ s a ) a and f ε ( s ) := f ( s ) + εs we have the following “radial”representations for f and f ε : (2.4) f ( q ) = f ( | q | ) q | q | and f ε ( q ) = f ε ( | q | ) q | q | for every q = . (iii) For ε > the function f ε is a diffeomorphism from R d onto R d , while f is a diffeomorphism from R d onto the open unit ball B (0) ⊂ R d .Proof. For q = we have ∂f εi ( q ) ∂q j = ∂∂q j (cid:18) q i (1 + | q | a ) a (cid:19) + εδ ij = (1 + | q | a ) δ ij − | q | a − q i q j (1 + | q | a ) a + εδ ij . This result can be easily extended to q = . Indeed, the above formula for partial derivatives is clearlycontinuous on R d \ { } and since a > | q i q j | ≤ | q | for all i, j ∈ { . . . , d } , we conclude | q | a − q i q j → q → . Thus f , f ε ∈ C ( R d ; R d ). This proves the first assertion.As the vectors q and f ε ( q ) have the same direction, the formulae (2.4) follow. Furthermore, lim s → + f ( s ) = 0,lim s →∞ f ( s ) = 1 and f ′ ( s ) = (1 + s a ) − aa >
0. Consequently, f is a strictly increasing C -function mapping[0 , ∞ ) onto [0 ,
1) and, for any ε > f ε is a strictly increasing C -function mapping [0 , ∞ ) onto [0 , ∞ ). Hencethe functions f − ( y ) := f − ( | y | ) y | y | and ( f ε ) − ( y ) := ( f ε ) − ( | y | ) y | y | are well defined inverse functions of f and f ε , respectively. It is straightforward to check that f − and ( f ε ) − are continuously differentiable, which completes the proof of (ii) and (iii). (cid:3) Next, we set(2.5) A ( q ) := ∇ q f ( q ) i.e. A ( q ) = (1 + | q | a ) I − | q | a − q ⊗ q (1 + | q | a ) a and we focus on its (finer) properties. (In (2.5), I stands for the identity matrix and ( q ⊗ q ) ij = q i q j .) Lemma 2.2 (Scalar product generated by ∇ q f ( q )) . Let q ∈ R d be arbitrary. The bilinear form on R d given by (2.6) ( v , w ) A ( q ) := v · A ( q ) w = d X i,j =1 v i ∂f i ( q ) ∂q j w j = (1 + | q | a ) v · w − | q | a − ( q · v )( q · w )(1 + | q | a ) a is a scalar product on R d satisfying (2.7) ( v , w ) A ( q ) ≤ | v | | w | for every v , w ∈ R d . The corresponding quadratic form v A ( q ) := ( v , v ) A ( q ) fulfills (2.8) | v | ≥ | v | (1 + | q | a ) a ≥ v A ( q ) ≥ | v | (1 + | q | a ) a for every v ∈ R d Hence, · A ( q ) is for fixed q ∈ R d the norm on R d equivalent to the Euclidean norm | · | .Proof. The proof follows from the definition of f , the formula (2.3) for its derivatives, (2.6) and the Cauchy-Schwarz inequality. The inequalities in (2.8) are direct consequences of (2.6). (cid:3) M. BUL´IˇCEK, D. HRUˇSKA, AND J. M´ALEK
The last essential property we need in the proof is the strict monotonicity of f , the strong monotonicity of f ε and, consequently, the Lipschitz continuity of its inverse function ( f ε ) − . Lemma 2.3.
The mappings f , f ε : R d → R d defined in (2.1) and (2.2) satisfy, for all ε ∈ (0 , , (cid:0) f ( q ) − f ( q ) (cid:1) · ( q − q ) > for all q , q ∈ R d , q = q , (2.9) (cid:0) f ε ( q ) − f ε ( q ) (cid:1) · ( q − q ) ≥ ε | q − q | for all q , q ∈ R d . (2.10) Moreover, for any ε > , the inverse function ( f ε ) − is uniformly Lipschitz continuous on R d , namely, (cid:12)(cid:12) ( f ε ) − ( y ) − ( f ε ) − ( y ) (cid:12)(cid:12) ≤ ε | y − y | for all y , y ∈ R d . (2.11) Proof.
We first observe, using also (2.5), that (for q = q ) (cid:0) f ε ( q ) − f ε ( q ) (cid:1) · ( q − q ) = Z dd s f ε ( q + s ( q − q )) d s · ( q − q )= Z A ( q + s ( q − q ))( q − q ) · ( q − q ) d s + ε | q − q | > ε | q − q | , which gives the strong monotonicity of f ε and strict monotonicity of f . Since (cid:0) f ε ( q ) − f ε ( q ) (cid:1) · ( q − q ) ≤ | f ε ( q ) − f ε ( q ) | | q − q | , we conclude from the last two inequalities that ε | q − q | ≤ | f ε ( q ) − f ε ( q ) | , which is equivalent to (2.11). (cid:3) Proof of uniqueness
In this short section, we shall prove that there is at most one weak solution to the problem (1.1).Let us assume that there are two weak solutions ( u , q ) and ( u , q ) to the problem (1.1) with the sameinitial value u ∈ L (Ω) and the same right-hand side g ∈ L ( Q ). Note that the constitutive equation (1.2b)implies that ∇ u , ∇ u ∈ L ∞ ( Q ) and consequently u and u are admissible test function in (1.2a). Subtracting(1.2a) for ( u , q ) from the same equation for ( u , q ) and taking ϕ = u ( t, · ) − u ( t, · ) as a test function, weobtain(3.1) Z Ω ( ∂ t u − ∂ t u )( u − u ) + ( q − q ) · ( ∇ u − ∇ u ) d x = 0 for a.a. t ∈ (0 , T ) . By (1.2b), ∇ u − ∇ u = f ( q ) − f ( q ). Inserting this relation into (3.1), we obtain12 dd t k u − u k L (Ω) + Z Ω ( f ( q ) − f ( q )) · ( q − q ) d x = 0 . Integrating this with respect to time t ∈ (0 , T ] and using u (0 , x ) − u (0 , x ) = 0 a.e. in Ω we arrive at12 k u ( t, · ) − u ( t, · ) k L (Ω) + Z t Z Ω ( f ( q ) − f ( q )) · ( q − q ) d x d s = 0 . By taking t = T and using the strict monotonicity of f , see (2.9), the second term leads to the conclusion that q = q a.e. in (0 , T ) × Ω. The first term then implies that, for all t ∈ (0 , T ], u ( t, · ) = u ( t, · ) a.e. in Ω. Thiscompletes the proof of uniqueness.4. ε -approximations and their properties In this section, we introduce, for any ε ∈ (0 , ε -approximation of the problem (1.1) and show, by meansof the Galerkin method and regularity techniques performed at the Galerkin level, that this ε -approximationadmits a unique weak solution with second spatial derivatives in L ( Q ).Let ε ∈ (0 ,
1) and a >
0. We say that a couple of Ω-periodic functions ( u, q ) = ( u ε , q ε ) solves the ε -approximation of the problem (1.1) if ∂ t u − div q = g in Q, (4.1a) ∇ u = q (1 + | q | a ) a + ε q = f ( q ) + ε q = f ε ( q ) in Q, (4.1b) u (0 , · ) = u in Ω . (4.1c) N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 9
In accordance with the assumptions of Theorem 1.2, we assume that u ∈ W , ∞ per (Ω) satisfies (1.3) and g ∈ L ( Q ). We say that a couple ( u, q ) = ( u ε , q ε ) is weak solution to (4.1) if(4.2) u ∈ L (cid:0) , T ; W , per (Ω) (cid:1) ,∂ t u ∈ L (cid:0) , T ; L (Ω) (cid:1) , q ∈ L (cid:0) , T ; L (cid:0) Ω; R d (cid:1)(cid:1) and Z Ω ∂ t u ϕ + q · ∇ ϕ d x = Z Ω g ϕ d x for all ϕ ∈ W , per (Ω) and a.a. t ∈ (0 , T ) , (4.3a) ∇ u = f ε ( q ) a.e. in Q, (4.3b) k u ( t, · ) − u k L (Ω) t → + −−−−→ . (4.3c)Uniqueness of such a solution follows from the same argument as in Section 3. To establish the existence of thesolution, we apply the Galerkin method combined with higher differentiability estimates that we will performat the level of Galerkin approximations. These estimates and the limit from the Galerkin approximation to thecontinuous level represent the core of this section. In Subsect. 4.6, we establish and summarize the estimatesthat are uniform with respect to ε .4.1. Galerkin approximations.
Consider the basis { ω r } ∞ r =1 in W , per (Ω) consisting of solutions of the followingspectral problem: Z Ω ∇ ω r · ∇ ϕ d x = λ r Z Ω ω r ϕ d x for all ϕ ∈ W , per (Ω) . (4.4)It is well-known (see e.g. [28] or [24, Appendix A.4]) that there is a non-decreasing sequence of (positive)eigenvalues { λ r } ∞ r =1 and a corresponding set of eigenfunctions { ω r } ∞ r =1 that are orthogonal in W , per (Ω) andorthonormal in L per (Ω). Moreover, the projections P N defined through P N ( u ) = P Ni =1 (cid:0)R Ω uω i d x (cid:1) ω i arecontinuous both as mappings from L per (Ω) to L per (Ω) and from W , per (Ω) to W , per (Ω). Also, due to Ω-periodicityand elliptic regularity, the Ω-periodic extensions of ω r belong to C ∞ ( R d ).Before introducing the Galerkin approximations of the problem (4.3) we recall, referring to Lemma 2.1, thatthe relation ∇ u = f ε ( q ) is equivalent to q = ( f ε ) − ( ∇ u ) where ( f ε ) − is a Lipschitz mapping from R d to R d .For an arbitrary, fixed N ∈ N , we look for u N in the form u N ( t, x ) = N X r =1 c Nr ( t ) ω r ( x ) , where the coefficients c Nr , r = 1 , . . . , N , are determined as the solution of the system of ordinary differentialequations of the form Z Ω ∂ t u N ω r + q N · ∇ ω r d x = Z Ω g ω r d x, r = 1 , . . . , N, where q N := ( f ε ) − ( ∇ u N ) , (4.5a) u N (0 , · ) = P N ( u ) ⇐⇒ c Nr (0) = Z Ω u ω r d x r = 1 , . . . , N. (4.5b)The local-in-time well-posedness of the above problem (4.5) directly follows from Caratheodory theory (recallhere that ( f ε ) − is a Lipschitz mapping). In addition, thanks to the first uniform estimates established in thenext subsection, we deduce that the Galerkin system (4.5) is well-posed on (0 , T ].4.2. First uniform estimates.
Multiplying the r -th equation in (4.5a) by c r and summing these equationsup for r = 1 , . . . , N , we obtain 12 dd t (cid:13)(cid:13) u N (cid:13)(cid:13) L (Ω) + Z Ω q N · ∇ u N d x = Z Ω g u N d x. Using the one-to-one correspondence between q N and ∇ u N , see (4.5a), the second term on the left-hand sidecan be evaluated explicitly and the above equation takes the form12 dd t (cid:13)(cid:13) u N (cid:13)(cid:13) L (Ω) + Z Ω (cid:12)(cid:12) q N (cid:12)(cid:12) (cid:0) | q N | a (cid:1) a + ε (cid:12)(cid:12) q N (cid:12)(cid:12) d x = Z Ω g u N d x ≤ k g k L (Ω) + 12 (cid:13)(cid:13) u N (cid:13)(cid:13) L (Ω) . Integrating over time, using then the Gronwall inequality and the fact that kP N u k L (Ω) ≤ k u k L (Ω) , we obtain(4.6) sup t ∈ (0 ,T ) (cid:13)(cid:13) u N ( t, · ) (cid:13)(cid:13) L (Ω) + Z T Z Ω (cid:12)(cid:12) q N (cid:12)(cid:12) (cid:0) | q N | a (cid:1) a + ε (cid:12)(cid:12) q N (cid:12)(cid:12) d x d t ≤ C (cid:16) k u k L (Ω) , k g k L ( Q ) (cid:17) . In addition, it also directly follows from ∇ u N = f ε ( q N ) (see the second equation in (4.5a)) and the above L estimate on q N that(4.7) Z T Z Ω (cid:12)(cid:12) ∇ u N (cid:12)(cid:12) d x d t ≤ C (cid:16) k u k L (Ω) , k g k L ( Q ) (cid:17) . Time derivative estimate (uniform with respect to N ). Multiplying the r -th equation in (4.5a) by dd t c r and summing these equations up for r = 1 , . . . , N , we obtain Z Ω (cid:12)(cid:12) ∂ t u N (cid:12)(cid:12) + q N · ∂ t (cid:0) ∇ u N (cid:1) d x = Z Ω g ∂ t u N d x. Applying Young’s inequality to the term on the right-hand side, we get(4.8) Z Ω (cid:12)(cid:12) ∂ t u N (cid:12)(cid:12) + 2 q N · ∂ t (cid:0) ∇ u N (cid:1) d x ≤ Z Ω | g | d x. Next, we focus on the second term on the left-hand side. Since ∇ u N = f ε ( q N ), it follows from the definition of f ε that q N · ∂ t (cid:0) ∇ u N (cid:1) = ∂ t (cid:0) q N · ∇ u N (cid:1) − ∂ t q N · ∇ u N = ∂ t (cid:12)(cid:12) q N (cid:12)(cid:12) (cid:0) | q N | a (cid:1) a + ε (cid:12)(cid:12) q N (cid:12)(cid:12) − ∂ t q N · q N (cid:0) | q N | a (cid:1) a + ε q N = ε ∂ t (cid:16)(cid:12)(cid:12) q N (cid:12)(cid:12) (cid:17) + ∂ t (cid:12)(cid:12) q N (cid:12)(cid:12) (cid:0) | q N | a (cid:1) a − ∂ t (cid:0)(cid:12)(cid:12) q N (cid:12)(cid:12)(cid:1) (cid:12)(cid:12) q N (cid:12)(cid:12)(cid:0) | q N | a (cid:1) a = ε ∂ t (cid:16)(cid:12)(cid:12) q N (cid:12)(cid:12) (cid:17) + ∂ t Z | q N | (cid:12)(cid:12) q N (cid:12)(cid:12)(cid:0) | q N | a (cid:1) a − s (1 + s a ) a d s. Inserting the result of this computation into (4.8), integrating the result over (0 , T ), and using the fact that thefunction s s (1 + s a ) a is increasing (implying that (cid:12)(cid:12) q N (cid:12)(cid:12) (1 + (cid:12)(cid:12) q N (cid:12)(cid:12) a ) − a − s (1 + s a ) − a ≥ , (cid:12)(cid:12) q N (cid:12)(cid:12) ), we obtain that Z T Z Ω (cid:12)(cid:12) ∂ t u N (cid:12)(cid:12) d x d t ≤ Z T Z Ω | g | − q N · ∂ t (cid:0) ∇ u N (cid:1) d x d t = Z T Z Ω | g | d x d t − Z Ω ε (cid:12)(cid:12) q N ( t, x ) (cid:12)(cid:12) + 2 Z | q N ( t,x ) | (cid:12)(cid:12) q N ( t, x ) (cid:12)(cid:12)(cid:0) | q N ( t, x ) | a (cid:1) a − s (1 + s a ) a d s d x t = Tt =0 ≤ Z T Z Ω | g | d x d t + Z Ω ε (cid:12)(cid:12) q N (0 , x ) (cid:12)(cid:12) + 2 Z | q N (0 ,x ) | (cid:12)(cid:12) q N (0 , x ) (cid:12)(cid:12)(cid:0) | q N (0 , x ) | a (cid:1) a − s (1 + s a ) a d s d x. Noticing that (cid:12)(cid:12) q N (cid:12)(cid:12) (1 + (cid:12)(cid:12) q N (cid:12)(cid:12) a ) − a − s (1 + s a ) − a ≤ , (cid:12)(cid:12) q N (cid:12)(cid:12) ) we conclude that Z T Z Ω (cid:12)(cid:12) ∂ t u N (cid:12)(cid:12) d x d t ≤ k g k L ( Q ) + ε (cid:13)(cid:13) q N (0 , · ) (cid:13)(cid:13) L (Ω; R d ) + 2 (cid:13)(cid:13) q N (0 , · ) (cid:13)(cid:13) L (Ω; R d ) , (4.9)where(4.10) q N (0 , x ) = ( f ε ) − ( ∇P N ( u ( x ))) ⇐⇒ ∇P N ( u ) = q N (0 , · ) (cid:0) | q N (0 , · ) | a (cid:1) a + ε q N (0 , · ) . By symbols such as C (cid:16) k u k L (Ω) , k g k L ( Q ) (cid:17) we indicate the dependence of the finite upper bound on “relevant” parameters(i.e. u , g , a and auxiliary parameters introduced in the proof such as ε ). The value of this bound can change from line to line. N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 11
Consequently, | q N (0 , · ) | ≤ ε |∇P N ( u ) | , which implies that k q N (0 , · ) k L (Ω; R d ) ≤ | Ω | / k q N (0 , · ) k / L (Ω; R d ) ≤ ε | Ω | / k∇P N ( u ) k / L (Ω; R d ) . The fact that kP N ( u ) k W , per (Ω) ≤ k u k W , per (Ω) thus finally yields Z T Z Ω (cid:12)(cid:12) ∂ t u N (cid:12)(cid:12) d x d t ≤ C (cid:16) ε − , k g k L ( Q ) , k u k W , per (Ω) (cid:17) . (4.11)4.4. Spatial derivative estimates.
This time, we multiply the r th equation in (4.5a) by λ r c r and sum theobtained identities up for r = 1 , . . . , N . Since, due to (4.4) and the smoothness of ω r , λ r Z Ω ω r ϕ d x = Z Ω ∇ ω r · ∇ ϕ d x = − Z Ω ∆ ω r ϕ d x for all ϕ ∈ W , per (Ω) , we get Z Ω ∂ t ∇ u N · ∇ u N + ∇ q N · ∇ u N d x = − Z Ω g ∆ u N d x. Hence,(4.12) dd t (cid:13)(cid:13) ∇ u N (cid:13)(cid:13) L (Ω; R d ) + 2 Z Ω ∇ q N · ∇ u N d x = − Z Ω g ∆ u N d x ≤ k g k L (Ω) (cid:13)(cid:13) ∇ u N (cid:13)(cid:13) L (Ω; R d × d ) . Since ∇ u N = f ε ( q N ), recalling (2.5) we get ∇ u N = A ( q N ) ∇ q N + ε ∇ q N . Hence, by Lemma 2.2, we get(4.13) ∇ q N · ∇ u N = ∇ q N · A ( q N ) ∇ q N + ε |∇ q N | = ∇ q N A ( q N ) + ε |∇ q N | and also, by means of the Cauchy-Schwarz inequality and (2.8), (cid:12)(cid:12) ∇ u N (cid:12)(cid:12) = A ( q N ) ∇ q N · ∇ u N + ε ∇ q N · ∇ u N ≤ ∇ q N A ( q N ) ∇ u N A ( q N ) + ε |∇ q N | |∇ u N |≤ ∇ q N A ( q N ) |∇ u N | + ε |∇ q N | |∇ u N | , which, using ε < ε , implies that(4.14) |∇ u N | ≤ ∇ q N A ( q N ) + ε |∇ q N | ) . Incorporating (4.13) and (4.14) into (4.12), integrating the result with respect to time and using Young’sinequality and the continuity of P N in W , per (Ω), we arrive at estimates that are uniform with respect to both N and ε : sup t ∈ (0 ,T ) (cid:13)(cid:13) ∇ u N ( t, · ) (cid:13)(cid:13) L (Ω; R d ) + Z T Z Ω ∇ q N A ( q N ) + ε (cid:12)(cid:12) ∇ q N (cid:12)(cid:12) + (cid:12)(cid:12) ∇ u N (cid:12)(cid:12) d x d t ≤ C (cid:16) k g k L ( Q ) , k u k W , per (Ω) (cid:17) . (4.15)4.5. Limit N → ∞ . Due to the reflexivity and separability of the underlying function spaces and the Aubin-Lions compactness lemma, it follows from the estimates (4.6), (4.7), (4.11) and (4.15) that there is a subsequenceof (cid:8) ( u N , q N ) (cid:9) ∞ N =1 (which we do not relabel) such that u N ⇀ u weakly in L (cid:0) , T ; W , per (Ω) (cid:1) , (4.16a) ∂ t u N ⇀ ∂ t u weakly in L (cid:0) , T ; L (Ω) (cid:1) , (4.16b) u N → u strongly in L (cid:0) , T ; W , per (Ω) (cid:1) ∩ C (cid:0) [0 , T ]; L (Ω) (cid:1) , (4.16c) q N ⇀ q weakly in L (cid:16) , T ; W , per (cid:0) Ω; R d (cid:1) (cid:17) . (4.16d)Letting N → ∞ in (4.5), it is simple to conclude from the above convergence results that Z Ω ∂ t u ϕ + q · ∇ ϕ d x = Z Ω g ϕ d x for all ϕ ∈ W , per (Ω) and a.a. t ∈ (0 , T ] . (4.17)Since u N (0 , · ) = P N ( u ), P N ( u ) → u in L (Ω) and u ∈ C (cid:0) [0 , T ]; L (Ω) (cid:1) , we observe that (4.3c) holds. By virtue of (4.16c) there is a subsequence (that we again do not relabel) so that(4.18) ∇ u N N →∞ −−−−→ ∇ u a.e. in Q. As ( f ε ) − is (Lipschitz) continuous, it follows from the second equation in (4.5a) and (4.18) that q N = ( f ε ) − (cid:0) ∇ u N (cid:1) N →∞ −−−−→ ( f ε ) − ( ∇ u ) a.e. in Q. Since the weak limit in L ( Q ) coincides with the pointwise limit a.e. in Q (provided that these limits exist), weconclude that(4.19) ( f ε ) − ( ∇ u ) = q a.e. in Q = ⇒ ∇ u = f ε ( q ) a.e. in Q. Thus, the existence and uniqueness of a weak solution to the ε -approximation (4.1) in the sense of definition(4.3) is completed.In the next subsection, we establish and summarize the estimates associated with the ε -approximation (4.1)that are uniform with respect to ε .4.6. ε -independent estimates for ( u ε , q ε ) . Observing that u ε is an admissible test function in (4.17), we set ϕ = u ε in (4.17). Then, proceeding step by step as at the Galerkin level, we obtain(4.20) sup t ∈ (0 ,T ) k u ε ( t, · ) k L (Ω) + Z T Z Ω | q ε | (1 + | q ε | a ) a + ε | q ε | d x d t ≤ C (cid:16) k u k , k g k L ( Q ) (cid:17) . It is easy to conclude from the boundedness of the second term, by applying H¨older’s inequality, that(4.21) Z T Z Ω | q ε | d x d t ≤ C (cid:16) | Ω | , k u k , k g k L ( Q ) (cid:17) . Further estimates are obtained by taking the limit N → ∞ in the estimates obtained at the Galerkin level.We define q (0 , · ) through the equation(4.22) ∇ u = f ε ( q (0 , · )) = q (0 , · )(1 + | q (0 , · ) | a ) /a + ε q (0 , · ) . As ∇P N ( u ) = f ε ( q N (0 , · )), see (4.10), ∇P N ( u ) → ∇ u strongly in L (Ω; R d ), and ( f ε ) − is Lipschitzcontinuous, we conclude that q N (0 , · ) N →∞ −−−−→ q (0 , · ) strongly in L (Ω; R d ) . Consequently, we can take the limit N → ∞ in (4.9) and conclude, using also the weak lower semicontinuity ofthe L -norm together with (4.16b), that Z T Z Ω | ∂ t u ε | d x d t ≤ k g k L ( Q ) + ε k q (0 , · ) k L (Ω; R d ) + 2 k q (0 , · ) k L (Ω; R d ) . (4.23)It follows from (1.3) and (4.22) that U ≥ |∇ u | = (cid:18) | q (0 , · ) | a ) a + ε (cid:19) | q (0 , · ) | ≥ | q (0 , · ) | (1 + | q (0 , · ) | a ) a a.e. in Q. This implies that | q (0 , · ) | ≤ U (1 − U a ) a . As U < k q (0 , · ) k L (Ω; R d ) ≤ C ( a, U ) and k q (0 , · ) k L (Ω; R d ) ≤ C ( a, U ) . The bound C ( a, U ) → ∞ as a →
0+ or as U → − . Inserting (4.24) into (4.23), we get(4.25) Z T Z Ω | ∂ t u ε | d x d t ≤ C ( a, U, k g k L ( Q ) ) . Finally, we let N → ∞ in (4.15). Recalling (4.16d) and also (4.18) together with (4.19), we have ∇ q N ⇀ ∇ q weakly in L ( Q ; R d × d ) , q N → q a.e. in Q. This implies (see the next subsection for the proof in a slightly more general setting) that Z T Z Ω ∇ q A ( q ) d x d t ≤ lim inf N →∞ Z T Z Ω ∇ q N A ( q N ) d x d t. N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 13
Consequently, letting N → ∞ in (4.15), we getsup t ∈ (0 ,T ) k∇ u ε ( t, · ) k L (Ω; R d ) + Z T Z Ω ∇ q ε A ( q ε ) + ε |∇ q ε | + (cid:12)(cid:12) ∇ u ε (cid:12)(cid:12) d x d t ≤ C (cid:16) k g k L ( Q ) , k u k W , per (Ω) (cid:17) . (4.26)4.7. Weak lower semicontinuity of the weighted L -norm. Here, we shall prove the following statement:if z n ⇀ z weakly in L ( Q ; R d ) as n → ∞ , (4.27) q n → q a.e. in Q as n → ∞ , (4.28)then(4.29) Z Q z A ( q ) d x d t ≤ lim inf n →∞ Z Q z n A ( q n ) d x d t. To prove it, we first recall that z A ( q ) = z · A ( q ) z , where A is introduced in (2.5). Observing that0 ≤ z n − z A ( q n ) = z n A ( q n ) − z A ( q n ) − z , z n − z ) A ( q n ) , we get(4.30) Z Q z n A ( q n ) d x d t ≥ Z Q z A ( q n ) d x d t + 2 Z Q ( z , z n − z ) A ( q n ) d x d t. Since | A ( q n ) | ≤ C ( d ) and (4.28) holds, Lebesgue’s dominated convergence theorem implies that(4.31) lim n →∞ Z Q z A ( q n ) d x d t = lim n →∞ Z Q z · A ( q n ) z d x d t = Z Q z A ( q ) d x d t. Furthermore, noticing that Z Q ( z , z n − z ) A ( q n ) d x d t = Z Q z · ( A ( q n ) − A ( q ))( z n − z ) d x d t + Z Q z · A ( q )( z n − z ) d x d t =: I n + I n , (4.32)we see that, as n → ∞ , I n vanishes by virtue of (4.27). To conclude that I n vanishes as well, we first applyH¨older’s inequality to get that | I n | ≤ k z n − z k L ( Q ; R d ) (cid:18)Z Q | z | | A ( q n ) − A ( q ) | d x d t (cid:19) / , and then we notice that k z n − z k L ( Q ; R d ) is bounded due to (4.27) and the last integral vanishes again byLebesgue’s dominated convergence theorem. Thus, lim n →∞ ( I n + I n ) = 0 and the assertion (4.29) follows from(4.30)-(4.32). 5. Limit ε → + The attainment of ∇ u = f ( q ) a.e. in Q . In Sect. 4, assuming that u ∈ W , ∞ per (Ω) satisfies (1.3) and g ∈ L ( Q ), we established, for any a > ε ∈ (0 , { ( u ε , q ε ) } ε ∈ (0 , satisfies the estimates (4.20),(4.21), (4.25) and (4.26). As a consequence of these estimates (that are uniform w.r.t. ε ) and the Aubin-Lionscompactness lemma, one can find ε m → u m , q m ) := ( u ε m , q ε m ) such that u m ⇀ u weakly in L (cid:0) , T ; W , per (Ω) (cid:1) , (5.1a) ∂ t u m ⇀ ∂ t u weakly in L (cid:0) , T ; L (Ω) (cid:1) , (5.1b) ∇ u m → ∇ u strongly in L (cid:0) , T ; L per (Ω; R d ) (cid:1) , (5.1c) ∇ u m → ∇ u a.e. in Q, (5.1d)and also, using (5.1d) and Egoroff’s theorem on one side and (4.21) and Chacon’s biting lemma (see [5]) onthe other side, there is a q ∈ L ( Q ; R d ) such that for each δ > Q δ ⊂ Q fulfilling ˜ Q δ ⊂ ˜ Q δ if δ ≤ δ as well as | Q \ ˜ Q δ | ≤ δ such that(5.2) q m ⇀ q weakly in L ( ˜ Q δ ; R d ) , ∇ u m → ∇ u strongly in L ∞ ( ˜ Q δ ; R d ) . Further, we denote Q δ := ˜ Q δ ∩ (cid:26) ( t, x ) ∈ Q ; | q ( t, x ) | ≤ δ (cid:27) and it follows from (5.2) that | Q \ Q δ | ≤ | Q \ ˜ Q δ | + | (cid:8) ( t, x ) ∈ Q ; | q ( t, x ) | > δ − (cid:9) | ≤ δ (cid:18) Z Q | q | d x d t (cid:19) ≤ Cδ.
Hence, using the (strict) monotonicity of f , see Lemma 2.3, the facts that f ( q ) ∈ L ∞ ( Q ; R d ) and f ( q m ) = ∇ u m − ε m q m , see (4.3b), the convergence properties (5.2), the obvious relation Q δ ⊂ ˜ Q δ , and the fact that q is bounded (depending on δ ) on Q δ , we observe that0 ≤ lim sup m →∞ Z Q δ (cid:0) f (cid:0) q m (cid:1) − f ( q ) (cid:1) · ( q m − q ) d x d t = lim sup m →∞ Z Q δ f ( q m ) · ( q m − q ) d x d t = lim sup m →∞ Z Q δ ∇ u m · ( q m − q ) − ε m q m · ( q m − q ) d x d t = lim sup m →∞ Z Q δ ( ∇ u m − ∇ u ) · ( q m − q ) + ∇ u · ( q m − q ) − ε m | q m | + ε m q m · q d x d t ≤ . This implies that there is a subsequence (that we again denote by q m ) such thatlim m →∞ (cid:0) f (cid:0) q m (cid:1) − f ( q ) (cid:1) · ( q m − q ) = 0 a.e. in Q δ . As f is strictly monotone, we conclude (referring for example to Lemma 6 in [14]) that q m → q a.e. in Q δ . However, as δ > | Q \ Q δ | ≤ Cδ , this yields(5.3) q m → q a.e. in Q. As f is continuous, letting m → ∞ in f ( q m ) = ∇ u m − ε m q m (valid a.e. in Q ) and using (5.1d) and (5.3), weconclude that (1.2b) holds.5.2. Limit in the governing evolutionary equation.
It remains to show that (1.2a) holds. Towards thisgoal, we “renormalize” the equation (4.3a) for ε m -approximation with the help of smooth, compactly supportedapproximations of unity denoted by τ k , which are the functions of | q m | . The required equation (1.2a) is thenobtained by a careful study of the limiting process as m → ∞ and k → ∞ .It follows from (4.3a) that, for all m ∈ N , Z Q ∂ t u m ϕ + q m · ∇ ϕ d x d t = Z Q g ϕ d x d t for all ϕ ∈ L (cid:0) , T ; W , per (Ω) (cid:1) . (5.4)In order to make use of these relations in the absence of weak convergence of q m in L ( Q ), we consider ψ ∈ L ∞ (cid:0) , T ; W , ∞ per (Ω) (cid:1) , and set as a test function ϕ in (5.4)(5.5) ϕ := τ k ( | q m | ) ψ, where τ k , k ∈ N , “approximates unity”, i.e. τ k ∈ C ∞ ([0 , ∞ )) satisfies for all k ∈ N the following conditions:0 ≤ τ k ( s ) ≤ s ∈ [0 , ∞ ), τ k ( s ) = 1 on [0 , k ], τ k ( s ) = 0 on [ k + 1 , ∞ ) and − ≤ τ ′ k ( s ) ≤ s ∈ ( k, k + 1). Note that, for fixed m , the test function specified in (5.5) is an admissible test function due to(4.26).Inserting (5.5) into (5.4), we obtain Z Q ∂ t u m τ k ( | q m | ) ψ d x d t + Z Q q m τ k ( | q m | ) · ∇ ψ d x d t = Z Q g τ k ( | q m | ) ψ d x d t − Z Q q m · ∇ τ k ( | q m | ) ψ d x d t. (5.6)Letting m → ∞ and k → ∞ in (5.6), our aim is to show that we can remove m and replace τ k by 1 in the firstthree integrals, while the last integral vanishes. Each term requires a special treatment.To treat the term involving the time derivative, we first observe that I m,k : = Z Q ( ∂ t u m τ k ( | q m | ) ψ − ∂ t uψ ) d x d t = Z Q ∂ t u m ( τ k ( | q m | ) − τ k ( | q | )) ψ d x d t + Z Q ( ∂ t u m − ∂ t u ) τ k ( | q | ) ψ d x d t − Z Q ∂ t u (1 − τ k ( | q | )) ψ d x d t =: J m,k + J m,k − J m,k . (5.7) N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 15
By H¨older’s inequality, (4.25), (5.3) and Lebesgue’s dominated convergence theorem, we observe that | J m,k | ≤ k ∂ t u m k L ( Q ) k ψ k L ∞ ( Q ) (cid:18)Z Q | τ k ( | q m | ) − τ k ( | q | ) | d x d t (cid:19) / → m → ∞ . By (5.1b), J m,k → m → ∞ . Using Levi’s monotone convergence theorem we also get | J m,k | ≤ Z Q | ∂ t u | | ψ | (1 − τ k ( | q | )) d x d t → k → ∞ . Consequently, it follows from (5.7) and the above arguments thatlim k →∞ lim m →∞ Z Q ∂ t u m τ k ( | q m | ) ψ d x d t = Z Q ∂ t u ψ d x d t. Even simpler arguments give(5.8) lim k →∞ lim m →∞ Z Q g τ k ( | q m | ) ψ d x d t = Z Q g ψ d x d t. Since, by (5.3) and Lebesgue’s dominated convergence theorem, (cid:12)(cid:12)(cid:12)(cid:12)Z Q ( q m τ k ( | q m | ) − q τ k ( | q | )) · ∇ ψ d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ ψ k L ∞ ( Q ) Z Q | q m τ k ( | q m | ) − q τ k ( | q | ) | d x d t → m → ∞ , we also observe (again using Levi’s monotone convergence theorem) thatlim k →∞ lim m →∞ Z Q q m τ k ( | q m | ) · ∇ ψ d x d t = Z Q q · ∇ ψ d x d t. It remains to show that the last term in (5.6) tends to zero as m, k → ∞ . To prove this, we will incorporate theweighted L -estimates for ∇ q m , see (4.26). Before starting to treat this term, we introduce, for every k ∈ N ,an auxiliary function G k through G k ( t ) := Z t τ ′ k ( s )(1 + s a ) a d s for t ∈ [0 , ∞ )and observe that G k ( t ) = 0 on [0 , k ] and(5.9) | G k ( t ) | ≤ Z k +1 k | τ ′ k ( s ) | a s d s ≤ a +1 a (1 + k ) ≤ C ( a ) t for every t ≥ k. Let us now rewrite the last term in (5.6) in the following manner: Z Q ψ q m · ∇ τ k ( | q m | ) d x d t = Z Q ψ q m · ∇ ( | q m | )(1 + | q m | a ) a τ ′ k ( | q m | ) (1 + | q m | a ) a d x d t = Z Q ψ f ( q m ) · ∇ G k ( | q m | ) d x d t = − Z Q ∇ ψ · f ( q m ) G k ( | q m | ) d x d t − Z Q ψG k ( | q m | ) div f ( q m ) d x d t =: J m,k + J m,k . (5.10)To show that J m,k vanishes as m → ∞ and k → ∞ , we first employ, for any fixed k , (5.3) and Lebesgue’sdominated convergence theorem (noticing that |∇ ψ · f ( q m ) | G k ( | q m | ) ≤ a +1 a (1 + k ) k∇ ψ k L ∞ ( Q ; R d ) ) and obtainlim m →∞ Z Q ∇ ψ · f ( q m ) G k ( | q m | ) d x d t → Z Q ∇ ψ · f ( q ) G k ( | q | ) d x d t. Since G k ( t ) = 0 on [0 , k ], we conclude from the estimate (5.9) and the fact that q ∈ L ( Q ; R d ) that (cid:12)(cid:12)(cid:12)(cid:12)Z Q ∇ ψ · f ( q ) G k ( | q | ) d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Q ∩{| q | >k } |∇ ψ · f ( q ) G k ( | q | ) | d x d t ≤ C ( a ) k∇ ψ k L ∞ ( Q ; R d ) Z Q ∩{| q | >k } | q | d x d t k →∞ −−−−→ . Hence, lim k →∞ lim m →∞ J m,k = 0.In order to show that also the term J m,k vanishes as m → ∞ and k → ∞ we need to proceed more carefully.First, recalling (2.5), we notice thatdiv ( f ( q m )) = ( A ( q m )) ij ∂ i q mj = ( A ( q m )) ij ∂ s q mj δ is = d X s =1 ( ∂ s q m , e s ) A ( q m ) a.e. in Q, where e s ∈ R d is the s th vector of the canonical basis in R d , s = 1 , . . . , d . This allows us to rewrite and estimate J m,k introduced in (5.10) as follows: (cid:12)(cid:12)(cid:12) J m,k (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Q d X s =1 ( ∂ s q m , ψG k ( | q m | ) e s ) A ( q m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x d t ≤ k ψ k L ∞ ( Q ) d X s =1 Z Q ∂ s q m A ( q m ) G k ( | q m | ) e s A ( q m ) d x d t ≤ k ψ k L ∞ ( Q ) d X s =1 (cid:18)Z Q ∇ q m A ( q m ) d x d t (cid:19) (cid:18)Z Q G k ( | q m | ) e s A ( q m ) d x d t (cid:19) (4.26) ≤ C (cid:16) k g k L ( Q ) , k u k W , per (Ω) (cid:17) k ψ k L ∞ ( Q ) d X s =1 (cid:18)Z Q G k ( | q m | ) e s A ( q m ) d x d t (cid:19) (2.8) ≤ C (cid:16) d, k g k L ( Q ) , k u k W , per (Ω) (cid:17) k ψ k L ∞ ( Q ) (cid:18)Z Q | G k ( | q m | ) | (1 + | q m | a ) /a d x d t (cid:19) . Letting m → ∞ in the last term, using (5.3), (5.9) and Lebesgue’s dominated convergence theorem, we get(5.11) lim sup m →∞ (cid:12)(cid:12)(cid:12) J m,k (cid:12)(cid:12)(cid:12) ≤ C (cid:16) d, k g k L ( Q ) , k u k W , per (Ω) (cid:17) k ψ k L ∞ ( Q ) (cid:18)Z Q | G k ( | q | ) | (1 + | q | a ) /a d x d t (cid:19) . However, as G k ( s ) = 0 on [0 , k ] and (5.9) holds, we further observe that Z Q G k ( | q | )(1 + | q | a ) /a d x d t = Z Q ∩{| q | >k } G k ( | q | )(1 + | q | a ) /a d x d t ≤ C ( a ) Z Q ∩{| q | >k } | q | | q | d x d t ≤ C ( a ) Z Q ∩{| q | >k } | q | d x d t → k → ∞ . Hence lim k →∞ lim m →∞ (cid:12)(cid:12)(cid:12) J m,k (cid:12)(cid:12)(cid:12) = 0 and, taking all computations starting at (5.10) into consideration, the lastterm in (5.6) vanishes as m → ∞ and k → ∞ . The proof of the first part of Theorem 1.2 is complete.6. Improved time derivative estimates and higher integrability of the flux for a ∈ (0 , / ( d + 1))In order to prove the second part of Theorem 1.2, we will combine the uniform spatial derivative estimatesestablished in (4.26) for ( u ε , q ε ) together with the uniform time derivative estimates that we are going to provenext.6.1. Improved time derivative estimates.
Consider, for any ε ∈ (0 , u ε , q ε )to (4.1) satisfying (4.2) and (4.3). It follows from (4.3a), (4.3c) and the assumption u ∈ W , per (Ω) that, for τ ∈ (0 , T ],(6.1) Z Ω ( u ε ( τ, · ) − u ) u d x + Z τ Z Ω q ε · ∇ u d x d s = Z τ Z Ω g u d x d s. By setting ϕ = u ε in (4.3a), followed by integration over time between 0 and τ , we also have(6.2) k u ε ( τ, · ) k L (Ω) − k u k L (Ω) + 2 Z τ Z Ω q ε · ∇ u ε d x d s = 2 Z τ Z Ω g u ε d x d s. Step 1.
For any z : [0 , T ] × Ω → R and for τ ∈ R such that t + τ ∈ [0 , T ], we set δ τ z ( t, x ) := z ( t + τ, x ) − z ( t, x ) τ . Taking the weak formulation (4.3a) at t + τ , followed by subtracting (4.3a) at t , and taking then ϕ = τ δ τ u ε asa test function in the resulting equation, we obtain(6.3) 12 dd t k δ τ u ε k L (Ω) + Z Ω δ τ q ε · ∇ δ τ u ε d x = Z Ω δ τ g δ τ u ε d x. Using (4.3b) (or (4.1b)) and (2.5), we observe that δ τ q ε · ∇ δ τ u ε = δ τ q ε · δ τ ∇ u ε = Z δ τ q ε · A ( q εθ,τ ) δ τ q ε d θ + ε | δ τ q ε | = Z δ τ q ε A ( q εθ,τ ) d θ + ε | δ τ q ε | , (6.4)where q εθ,τ ( t, x ) := q ε ( t, x ) + θ ( q ε ( t + τ, x ) − q ε ( t, x )) for θ ∈ (0 , . N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 17
Inserting (6.4) into (6.3) and using the Cauchy-Schwarz inequality to estimate the right-hand side and theGronwall lemma, we conclude that for a.a. t ∈ (0 , T ] the following estimates holds:(6.5) k δ τ u ε ( t, · ) k L (Ω) + Z t Z Ω Z δ τ q ε A ( q εθ,τ ) d θ + ε | δ τ q ε | d x d s ≤ C e T k δ τ u ε (0 , · ) k L (Ω) + Z T k δ τ g k L (Ω) d s ! . This would lead to the required ε -independent estimates provided that we can control k δ τ u ε (0 , · ) k L (Ω) uniformlyw.r.t. ε . Step 2.
Towards this aim, we start by noticing that trivially k δ τ u ε (0 , · ) k L (Ω) = 1 τ k u ε ( τ, · ) − u k L (Ω) and(6.6) k u ε ( τ, · ) − u k L (Ω) = k u ε ( τ, · ) k L (Ω) − k u k L (Ω) − Z Ω ( u ε ( τ, · ) − u ) u d x. Inserting (6.1) and (6.2) into (6.6) we get12 k u ε ( τ, · ) − u k L (Ω) = Z τ Z Ω gu ε d x d s − Z τ Z Ω q ε · ∇ u ε d x d s − Z τ Z Ω gu d x d s + Z τ Z Ω q ε · ∇ u d x d s. This can be rewritten as12 k u ε ( τ, · ) − u k L (Ω) + Z τ Z Ω ( q ε − q ε (0 , · )) · ( ∇ u ε − ∇ u ) d x d s = Z τ Z Ω ( g − g (0 , · )( u ε − u ) d x d s + Z τ Z Ω g (0 , · )( u ε − u ) d x d s + Z τ Z Ω div q ε (0 , · ) ( u ε − u ) d x d s, (6.7)where q ε (0 , · ) is defined, in accordance with Subsect. 4.6, through(6.8) ∇ u = f ε ( q ε (0 , · )) = q ε (0 , · )(1 + | q ε (0 , · ) | a ) /a + ε q ε (0 , · ) . Since ∇ u ε = f ε ( q ε ) and f ε is monotone, the second term at the left-hand side of (6.7) is nonnegative. Intro-ducing the notation y ( τ ) := 12 Z τ k u ε ( s, · ) − u k L (Ω) d s and A ( s ) := k g ( s, · ) − g (0 , · ) k L (Ω) + k g (0 , · ) k L (Ω) + k∇ q ε (0 , · ) k L (Ω; R d × d ) , we conclude from (6.7), using H¨older’s inequality, that(6.9) y ′ ( τ ) = 12 k u ε ( τ, · ) − u k L (Ω) d s ≤ Z τ A ( s ) k u ε ( s, · ) − u k L (Ω) d s ≤ (cid:18)Z τ A ( s ) d s (cid:19) / y ( τ ) / . This (together with relabelling s on v and τ on s ) implies that(6.10) ( y / ) ′ ( s ) ≤ (cid:18)Z s A ( v ) d v (cid:19) / . Since y (0) = 0, integrating (6.10) over (0 , τ ) and using then H¨older’s inequality, we get y ( τ ) ≤ Z τ (cid:18)Z s A ( v ) d v (cid:19) / d s ! ≤ τ Z τ Z s A ( v ) d v d s ≤ τ Z τ A ( s ) d s, which implies that ( y ( τ )) / ≤ τ (cid:18)Z τ A ( s ) d s (cid:19) / . Using this to estimate the last term in (6.9), it follows from (6.9) that(6.11) k u ε ( τ, · ) − u k L (Ω) d s ≤ τ Z τ A ( s ) d s. Recalling the definition of A , (6.11) leads to (using also 1 /τ ≤ /s ) k u ε ( τ, · ) − u k L (Ω) ≤ Cτ Z τ k g ( s, · ) − g (0 , · ) k L (Ω) + k g (0 , · ) k L (Ω) + k∇ q ε (0 , · ) k L (Ω; R d × d ) d s ≤ Cτ (cid:18) k∇ q ε (0 , · ) k L (Ω; R d × d ) + k g (0 , · ) k L (Ω) + τ Z τ k δ s g (0 , · ) k L (Ω) d s (cid:19) . This finally gives k δ τ u ε (0 , · ) k L (Ω) ≤ C (cid:18) k∇ q ε (0 , · ) k L (Ω; R d × d ) + k g (0 , · ) k L (Ω) + τ Z τ k δ s g (0 , · ) k L (Ω) d s (cid:19) . As g ∈ W , (cid:0) , T ; L (Ω) (cid:1) and W , (cid:0) , T ; L (Ω) (cid:1) ֒ → C ([0 , T ]; L (Ω)), the second and the third terms on theright-hand side are bounded. Hence, we finally get(6.12) k δ τ u ε (0 , · ) k L (Ω) ≤ C (cid:0) k g k W , (0 ,T ; L (Ω)) (cid:1) + C k∇ q ε (0 , · ) k L (Ω; R d × d ) . In order to estimate k∇ q ε (0 , · ) k L (Ω; R d × d ) , we first recall that it follows from (1.3) and (6.8) that U ≥ |∇ u | = (cid:18) | q ε (0 , · ) | a ) a + ε (cid:19) | q ε (0 , · ) | ≥ | q ε (0 , · ) | (1 + | q ε (0 , · ) | a ) a a.e. in Q, which implies that(6.13) | q ε (0 , · ) | a ≤ U a − U a and (1 + | q ε (0 , · ) | a ) a ≤ − U a ) a . Next, applying the partial derivative w.r.t. x j to (6.8) and using also (2.5) we get ∇ ∂ j u = A ( q ε (0 , · )) ∂ j q ε (0 , · ) + ε∂ j q ε (0 , · ) . Taking the scalar product of this identity with ∂ j q ε (0 , · ) and summing the result over j , j = 1 , . . . , d , we arriveat ε |∇ q ε (0 , · ) | + ∇ q ε (0 , · ) A ( q ε (0 , · )) = ∇ u · ∇ q ε (0 , · ) ≤ |∇ u | |∇ q ε (0 , · ) | . By virtue of (2.8), this leads to |∇ q ε (0 , · ) | (1 + | q ε (0 , · ) | a ) a ≤ |∇ u | |∇ q ε (0 , · ) | = ⇒ |∇ q ε (0 , · ) | ≤ |∇ u | (1+ | q ε (0 , · ) | a ) a (6.13) ≤ |∇ u | − U a ) a , which implies that k∇ q ε (0 , · ) k L (Ω; R d × d ) ≤ − U a ) a k∇ u k L (Ω; R d × d ) . Consequently, using (6.5) and (6.12), we conclude that(6.14) k δ τ u ε ( t, · ) k L (Ω) + Z t Z Ω Z δ τ q ε A ( q εθ,τ ) d θ + ε | δ τ q ε | d x d s ≤ C (cid:0) k g k W , (0 ,T ; L (Ω)) , k u k W , (Ω) , a, U (cid:1) . Step 3.
Letting τ → ε ∈ (0 ,
1) being fixed) we claim that(6.15) k ∂ t u ε ( t, · ) k L (Ω) + Z t Z Ω ∂ t q ε A ( q ε ) + ε | ∂ t q ε | d x d s ≤ C (cid:0) k g k W , (0 ,T ; L (Ω)) , k u k W , (Ω) , a, U (cid:1) . While the limits in the first and third terms of (6.14) are standard and are based on weak lower semicontinuityof the L -norm, the limit in the second term follows from the facts that, as τ → q εθ,τ → q ε a.e. in Q,δ τ q ε ⇀ ∂ t q ε weakly in L ( Q ; R d ) , followed by the convergence arguments established in Subsect. 4.7. Thus, (6.15) holds. Consequently, weconclude that ∂ t u ∈ L ∞ (0 , T ; L (Ω)), which is the first statement of part (ii) of Theorem 1.2. Note that it would be sufficient to assume that g ∈ W β, (cid:0) , T ; L (Ω) (cid:1) for some β > / N EVOLUTIONARY PROBLEMS WITH A-PRIORI BOUNDED GRADIENTS 19
Higher integrability result.
It follows from (4.26) and (6.15) that Z Q ∂ t q ε A ( q ε ) + ∇ q ε A ( q ε ) ≤ C (cid:0) k g k W , (0 ,T ; L (Ω)) , k u k W , (Ω) , U (cid:1) =: C ∗ . Introducing the time-spatial gradient ∇ t,x u := ( ∂ t u, ∂ j u, . . . , ∂ d u ), we can rewrite the last estimate as Z Q ∇ t,x q ε A ( q ε ) ≤ C ∗ . Using the last inequality in (2.8), it implies that Z Q |∇ t,x q ε | (1 + | q ε | ) a +1 d x d t ≤ C ∗ , and by simple manipulation also Z Q |∇ t,x (1 + | q ε | ) − a | d x d t ≤ C ∗ . Hence, using also (4.21), we conclude that, for a ∈ (0 , k (1 + | q ε | ) − a k W , ( Q ) ≤ C ∗ , and it then follows from Sobolev embedding that k (1 + | q ε | ) − a k L p ( Q ) ≤ C ∗ , where p < ∞ is arbitrary if d = 1 and p = d +1) d − if d >
1. Thus if d = 1 and a < d >
1, the above computation gives that Z Q (1 + | q ε | ) (1 − a )( d +1) d − d x d t ≤ C ∗ , which improves the integrability of { q ε } , uniformly w.r.t. ε , provided that(1 − a )( d + 1) d − > ⇔ d + 1 > a. For ε m →
0, this piece of information is preserved. Thus, the second assertion of Theorem 1.2 is established.7.
Generalization to systems of nonlinear parabolic equations
Finally, we generalize our problem and formulate the existence and uniqueness results for such a generaliza-tion. A detailed proof is not provided as it follows from the combination of the arguments developed in theproof of Theorem 1.2 above and from the arguments used when proving the result established in [8], where thestationary case is treated.
Theorem 7.1.
Let Ω , Q be as before and let F : R → R + be a strictly convex C , function fulfilling F (0) = 0 .Assume in addition that there exists a positive constant C such that for all s ∈ R there holds C − | s | − C ≤ F ( | s | ) ≤ C (1 + | s | ) . For N ∈ N arbitrary, set f ( q ) := ∂ q F ( | q | ) , where q ∈ R d × N . Let g ∈ L (cid:16) , T ; L (cid:0) Ω; R N (cid:1) (cid:17) , u ∈ W , per (cid:0) Ω; R N (cid:1) and there exist a compact set K ⊂ R d × N such that ∇ u ( x ) ∈ f ( K ) for a.a. x ∈ Ω . Then, there exists a unique couple ( u, q ) such that u ∈ W , (cid:16) , T ; L (cid:0) Ω; R N (cid:1) (cid:17) ∩ L (cid:16) , T ; W , per (cid:0) Ω; R N (cid:1) (cid:17) , q ∈ L (cid:16) , T ; L (cid:0) Ω; R d × N (cid:1) (cid:17) and Z Ω ∂ t u · ϕ + q · ∇ ϕ d x = Z Ω g · ϕ d x for all ϕ ∈ W , ∞ per (Ω; R N ) and a.a. t ∈ (0 , T ) , (7.1a) ∇ u = f ( q ) a.e. in Q, (7.1b) k u ( t, · ) − u k L (Ω; R N ) t → + −−−−→ . (7.1c) C ∗ is a generic constant, whose value can change from line to line. References [1] G. Akagi, P. Juutinen, and R. Kajikiya. Asymptotic behavior of viscosity solutions for a degenerate parabolic equationassociated with the infinity-laplacian.
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Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovsk´a 83, 186 75 Prague,Czech Republic
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