A Note on Measure-Valued Solutions to the Full Euler System
aa r X i v : . [ m a t h . A P ] S e p A Note on Measure-Valued Solutions to the Full Euler System
V´aclav M´acha ∗ Emil Wiedemann † Abstract
We construct two particular solutions of the full Euler system which emanate from the same initial data.Our aim is to show that the convex combination of these two solutions form a measure-valued solutionwhich may not be approximated by a sequence of weak solutions. As a result, the weak* closure of the setof all weak solutions, considered as parametrized measures, is not equal to the space of all measure-valuedsolutions. This is in stark contrast with the incompressible Euler equations.
In the context of fluid dynamics, measure-valued solutions were first studied by DiPerna and Majda [8], whodeveloped an appropriate mathematical framework and showed existence for such solutions of the incompress-ible Euler equations. Measure-valued solutions describe the one-point statistics of a fluid, i.e., they give theprobability distribution of the fluid velocity (and other state variables like density or temperature) at a givenpoint in time and space. If one is willing to accept such a probabilistic description rather than a deterministicone (which would, of course, contain more information), then one easily obtains a solution for any initial data,bypassing the notorious problem of non-interchangeability of weak limits and nonlinearities.Measure-valued solutions are sometimes thought of as a “cheap way out” of the fundamental lack of com-pactness for inviscid fluid models, and are criticized as not containing enough interesting information. Yet,in recent years, the concept has been intensely studied again, as it turned out to have several merits afterall: First, measure-valued solutions, despite representing a very weak notion of solution, enjoy a weak-strongstability property that entails important consequences for singular limits, numerical approximation, and long-time behaviour; this property is known for many systems of fluid mechanics, including the incompressible [2],isentropic compressible [13], and full compressible [4] Euler systems, the isentropic compressible Navier-Stokesequations [9], and the Navier-Stokes-Fourier equations [5], see also the survey [19]. One should remark thatweak-strong uniqueness holds only for admissible measure-valued solutions, which comply with appropriateenergy or entropy inequalities.Motivated by numerical simulations, Fjordholm et al. [10] argued that measure-valued solutions provide fora more suitable notion of solution than weak (distributional) solutions; indeed, the numerical computation ofunstable shear flows with randomly perturbed initial data yields highly unpredictable results on the level ofweak solutions, but apparently stable and regular behavior on the measure-valued level. This phenomenon isof course very plausible in the light of phenomenological turbulence theory [11].Measure-valued solutions seem like a vast generalization of weak solutions, but are they really? Surprisingly,the answer is ‘no’ for the incompressible Euler equations [18]. Indeed, any measure-valued solution is weaklyapproximated by a sequence of weak solutions (and if the measure is admissible, then the approximating sequencecan also be chosen to consist of admissible weak solutions), or in other words: The set of Dirac parametrizedmeasures is weakly* dense in the set of all measure-valued solutions. One might thus say that the notion ofmeasure-valued solution is just a topological closure, but not a substantial extension, of the more classicalconcept of weak solution.Looking at [18] from a different angle, one could view the result as an instance of a characterization of Youngmeasures generated by sequences with specific properties (viz., being a solution of the incompressible Eulerequations). The most classical result of this kind is the characterization of gradient Young measures [14, 15],where not every Young measure whose barycenter is a gradient is itself generated by a sequence of gradients.This already indicates that the situation for incompressible Euler is rather unusual.In fact, the fact that every measure-valued solution of the incompressible Euler equations is generable isrelated to the wave cone for the corresponding linear constraint; indeed, the wave cone in this case is the wholespace. It turns out that this is no longer the case for compressible models. In [7], the wave cone for the isentropiccompressible Euler system is determined (and it is not the whole space), and a preliminary application to thegenerability of measure-valued solutions is given. This result was recently extended in [12] to yield an admissible ∗ Institute of Mathematics of the Academy of Sciences of the Czech Republic, e-mail: [email protected] † Institute of Applied Analysis, Ulm University, Germany, e-mail: [email protected] PRELIMINARIES
Let
T >
0. We consider the following system on time-space [0 , T ] × R : ∂ t ρ + div x ( ρ v ) = 0 ∂ t ( ρ v ) + div x ( ρ v ⊗ v ) + ∇ p ( ̺, θ ) = 0 ∂ t (cid:18) ρ | v | + ρe ( ρ, θ ) (cid:19) + div x (cid:18)(cid:18) ρ | v | + ρe ( ρ, θ ) (cid:19) v (cid:19) + div x ( p ( ρ, θ ) v ) = 0 , (2.1)with unknowns ρ : [0 , T ] × Ω → R +0 , v = ( v, u ) : [0 , T ] × Ω → R and θ : [0 , T ] × Ω → R +0 . The functions e and p are interrelated through the Gibbs law which gives rise also to an entropy – a function s ( ρ, θ ) such that θDs ( ρ, θ ) = De ( ρ, θ ) + p ( ρ, θ ) D (cid:18) ρ (cid:19) , where D stands for the gradient with respect to ρ and θ . Throughout this paper we consider an ideal gas, i.e., p ( ρ, θ ) = ρθ, e ( ρ, θ ) = c v θ, s ( ρ, θ ) = log (cid:18) θ c v ρ (cid:19) , for some c v > c v = 1. We rewrite (2.1) into conservative variables ρ, m = ρ v , E = ρ | v | + ρe ( ρ, θ ). Our choice of state variables gives p = E − m ρ and thus we get ∂ t ρ + div x m = 0 ∂ t m + div x (cid:18) m ⊗ m ρ + I (cid:18) E − m ρ (cid:19)(cid:19) = 0 ∂ t E + div (cid:18)(cid:18) E − m ρ (cid:19) m ρ (cid:19) = 0 , (2.2)where I denotes the (2 ×
2) identity matrix. Furthermore, we may rewrite it as a linear differential system ∂ t ρ + div x m = 0 ∂ t m + div x U + ∇ E = 0 ∂ t E + div x r = 0 (2.3)with the following constraints: U = m ⊗ m ρ − m ρ I r = (cid:18)(cid:18) E − m ρ (cid:19) m ρ (cid:19) . (2.4) PRELIMINARIES U is a traceless symetric matrix and thus the system (2.3) maybe rewritten as div ρ m m m U + E U m U − U + EE r r = 0 . (2.5)Here div stands for a divergence over the time-space variables ( t, x ). Definition 2.1.
We say that a family of probability measures ν := ν t,x ∈ L ∞ w ∗ ([0 , T ] × R , P ( R +0 × R × R +0 )) is a measure-valued solution to (2.1) with initial data ( ρ , m , E ) if • Z T Z R ρ ( t, x ) ∂ t ϕ ( t, x ) + m ( t, x ) · ∇ ϕ ( t, x ) d x d t + Z R ρ ( x ) ϕ (0 , x )d x = 0 (2.6) for all ϕ ∈ C ∞ c ([0 , T ) × R ) , • Z T Z R m ( t, x ) · ∂ t ϕ ( t, x ) + (cid:28) ν t,x , ξ m ⊗ ξ m ξ ρ − I (cid:18) ξ E − ξ m ξ ρ (cid:19)(cid:29) : ∇ ϕ ( t, x ) d x d t + Z R m ( x ) · ϕ (0 , x )d x = 0(2.7) for all ϕ ∈ C ∞ c ([0 , T ) × R ) , • Z T Z R E ( t, x ) ∂ t ϕ ( t, x ) + (cid:28) ν t,x , (cid:18) ξ E − ξ m ξ ρ (cid:19) ξ m ξ ρ (cid:29) · ∇ ϕ ( t, x )d x d t + Z R E ( x ) ϕ (0 , x )d x = 0 (2.8) for all ϕ ∈ C ∞ c ([0 , T ) × R ) .Here ξ ρ ∈ R +0 , ξ m ∈ R , and ξ E ∈ R +0 are dummy variables for ρ , m and E , meaning that ρ ( t, x ) = Z ∞ ξ ρ d ν t,x , m ( t, x ) = Z R ξ m d ν t,x , E = Z ∞ ξ E d ν t,x and we use the notation h ν t,x , f ( ξ ρ , ξ m , ξ E ) i = Z R +0 × R × R +0 f ( ξ ρ , ξ m , ξ E ) d ν t,x . For the existence of a measure-valued solution, one needs an extended notion including concentration effects;if these are taken into account, then the existence of measure-valued solutions is known owing to J. Bˇrezina [3].His definition of measure-valued solution is also slightly different from ours in that he used the renormalizedentropy balance and the total energy balance instead of the energy balance. In any case, every weak solution( ρ, m , E ) is also a measure valued solution (in the sense of Definition 2.1) with ν t,x = δ ρ ( t,x ) ⊗ δ m ( t,x ) ⊗ δ E ( t,x ) .Thus, the existence of infinitely many weak solutions for certain initial data exhibited in [1] and in [16] showsa fortiori the existence of non-unique measure-valued solutions for these data.As an immediate consequence of the definition we obtain that every convex combination of two measurevalued solutions is also a measure-valued solution, that is, if ν, µ are two measure-valued solutions, then so is λν + (1 − λ ) µ for any λ ∈ [0 , z n ) : Ω → R d of measurable functions generatesthe parametrized measure ( ν x ) x ∈ Ω if f ( z n ) ⇀ h ν, f i weakly in L (Ω)for all continuous functions f : R d → R for which ( f ( z n )) is equi-integrable.We will take advantage of the following theorem proved in [7]: Theorem 2.2.
Let Ω ⊂ R N be a Lipschitz and bounded domain, ≤ p < ∞ , and A a linear operator of theform A z := N X i =1 A ( i ) ∂z∂x i , (2.9) where A ( i ) are l × d matrices and z : R N R d a vector valued function. Let p ∈ (1 , ∞ ) and z , z ∈ R d , z = z be two constant states such that z − z / ∈ Λ , PRELIMINARIES where Λ denotes the wave cone, defined by Λ = (cid:8) ¯ z ∈ R d : there exists ξ ∈ R N \ { } such that A (¯ zh ( · · ξ )) = 0 for all h : R → R (cid:9) . Let further z n : Ω → R d is an equi-integrable family of functions with k z n k L p ≤ c A z n → in W − ,r (Ω) for some r ∈ (cid:16) , NN − (cid:17) , and assume that ( z n ) generates a compactly supported Young measure such that supp[ ν x ] ⊂ { λz + (1 − λ ) z , λ ∈ [0 , } for a.a. x ∈ Ω . Then there exists z ∞ ∈ R d such that z n → z ∞ in L p (Ω) . In our setting (i.e., the state vector is ( ρ, m , m , U , U , E, r , r ), we have A (1) = ,A (2) = ,A (3) = − , For a given point z = ( ρ, m , m , U , U , E, r , r ) define a matrix ( Z A ) ji as follows:( Z A ) ji = d X k =1 A ( i ) jk z k , j = 1 , . . . , , i = 1 , . . . , . As observed in [7, Section 3.2], z ∈ Λ if and only if the corresponding Z A satisfies rank Z A <
3. Thus it isenough to take z = (cid:18) , , , , , , , (cid:19) , z = (cid:18) γ, , , γ , , γ , γ , (cid:19) . Trivially, both z and z are solutions to (2.2) since they are constant. Moreover, there exists γ such that z − z / ∈ Λ. Indeed, the corresponding Z A is of the form − γ (cid:16) − γ (cid:17)
00 0 1 − γ (1 − γ ) (1 − γ ) 0 and the determinant of the 3 × Z A ) i,j =1 is 2(1 − γ )(1 − γ ) and thus is nonzero for all γ = 1. Ac-cording to Theorem 2.2, the measure valued solution ν t,x = δ z + δ z may not be approximated by a sequenceof weak solutions. This is the cheapest way how to produce a solution of the demanded quality. However, theinitial datum for this solution is already a measure and one may ask, as we did in the introduction, whetherthere is a measure-valued solution emanating from deterministic initial data which cannot be approximated bya sequence of weak solutions. SOLUTION EMANATING FROM ‘ATOMIC’ INITIAL DATA We present a non-constant variation of Theorem 2.2 proven in [12]:
Theorem 3.1.
Let Ω ⊂ R be an open bounded domain, A a linear homogeneous constant rank differentialoperator of order one satisfying l ≥ , and ≤ p < ∞ . Further, let z , z ⊂ L ∞ (Ω , R m ) be such that z − z / ∈ Λ a.e. in Ω . Assume z n : Ω R m is an equi-integrable family of functions such that k z n k p ≤ c < ∞A z n → in W − ,r (Ω) , for some r ∈ (cid:16) , NN − (cid:17) , and { z n } generates a compactly supported Young measure ν ∈ L ∞ w (Ω , M ( R m )) suchthat supp( ν x ) ⊂ { λz ( x ) + (1 − λ ) z ( x ) , λ ∈ [0 , } for a.e. x ∈ Ω .Then, for a.e. x ∈ Ω it holds that ν x = δ w ( x ) with w ∈ L (Ω) and z n → w in L (Ω) . The question of the existence of the demanded measure-valued solution reduces to the question whetherthere are two weak solutions to (2.2) z and z emanating from the same “atomic” initial conditions such that z − z / ∈ Λ on a subset of Ω of positive measure. As noted before, we need to compute the rank of a certainmatrix.Let z = ( ρ α , m α , m α , U α , U α , E α , r α , r α ) and z = ( ρ β , m β , m β , U β , U β , E β , r β , r β ) be two solutions forwhich is the constraint (2.4) effective almost everywhere. The appropriate Z A is of the form ρ α − ρ β m α − m β m α − m β m α − m β U α − U β + E α − E β U α − U β m α − m β U α − U β U β − U α + E α − E β E α − E β r α − r β r α − r β . (3.1)Below we show the existence of two solutions ( ρ α , v α , p α ) and ( ρ β , v β , p β ) for which the appropriate matrix Z A is of rank 3 on a set of positive measure. Take Riemann initial data of the following form:( ρ, v , p ) = (cid:26) ( ρ − , ( v K , , p − ) for x < ρ K , (0 , , p + ) for x > , (3.2)where ρ K = ρ − p − + 3 p + p − + p + , v K = √ √ ρ − p + − p − √ p − + 3 p + and ρ − , p − , p + > p + > p − are given constants. According to [17] there is a self-similar solution consistingof a 1-shock. In particular, let s = − p + + 3 p − p ρ − ( p − + 3 p + ) . Then a triple ( ρ α , v α , p α ) = (cid:26) ( ρ − , ( v K , , p − ) for x < st, ( ρ K , (0 , , p + ) for x > st is a weak solution to (2.1), see Figure 1. Here we present the necessary details of the construction from [16]. First, according to [16, Section 4.3], wedefine a pressure p δ = p + + δ p and a velocity v δ = δ v where δ v = δ p q ρ K (4 p + +3 δ p ) . Also, set ρ δ = ρ K p δ + p + p + + p δ .Note that δ v = δ v ( δ p ) is a smooth function on a neighborhood of 0, δ v (0) = 0, and δ p is a positive arbitrarilysmall number. The time-space is then divided into regions Ω − , Ω , Ω , Ω δ and Ω + as shown in Figure 2. SOLUTION EMANATING FROM ‘ATOMIC’ INITIAL DATA x x = st ( ρ − , ( v K , , p − ) ( ρ K , (0 , , p + )Figure 1: Self-similar solution x Ω + Ω δ Ω Ω Ω − Figure 2: Fan partitionBetween Ω δ and Ω + there is a 3-shock. In order to handle the region Ω − ∪ Ω ∪ Ω ∪ Ω δ , we use the Galileantransformation ( ρ, ( v, u ) , p )( t, x ) ( ρ, ( v − δ v , u ) , p )( t, x + δ v t e )to get the right state ( ρ − , ( v K − δ v , , p − ) in Ω − and the left state ( ρ δ p , (0 , , p + + δ p ) in Ω δ . The Galilean transformation also changes the sets Ω − , Ω , Ω and Ω δ . However, we keep the same notation forthe sake of simplicity.According to [16, Theorem 3.1], there exist infinitely many admissible weak solutions to (2.1). We denoteone of these solutions by ( ρ β , ( v ′ ) β , p β ) – this solution has to be transformed back by Galilean transformationinto the solution ( ρ β , v β , p β ). Such a solution fulfills( ρ β , ( v ′ ) β , p β ) ↾ Ω − = ( ρ − , ( v K − δ v , , p − ) , ( ρ β , ( v ′ ) β , p β ) ↾ Ω δ = ( ρ δ , (0 , , p + + δ p ) . Moreover, we have the following: | ( v ′ ) β ↾ Ω | = ε + ˜ ε ρ β ↾ Ω = ρ ∈ R + ρ β ↾ Ω = ρ ∈ R + p β ↾ Ω = p p β ↾ Ω = p where ε = ε ( v − − δ p , ρ , ρ , p − , p + )˜ ε = ˜ ε ( v − − δ p , ρ , ρ , p − , p + ) p := p ( ρ , p − ) EFERENCES ρ K − ρ and ρ − ρ K may be arbitrarilysmall. As δ v and δ p are also arbitrarily small, we verify the property that the matrix (3.1) has full rank for ρ = ρ = ρ K , v − = v K and p δ = p + on a set of positive measure. The aforementioned continuity then allowsto use the intended approximation.The interface between Ω and Ω δ is { x = δ v t } . Consequently, the domain Ω ∩ { x > st } is nonempty andhas positive measure since s <
0. On this set we take z α = (cid:18) ρ α , ρ α v α , ρ α u α , ρ α (( v α ) − ( u α ) ) , ρ α u α v α , ρ α | v α | + p α , (cid:18) ρ α | v α | + 2 p α (cid:19) v α , (cid:18) ρ α | v α | + 2 p α (cid:19) u α (cid:19) z β = (cid:18) ρ β , ρ β v β , ρ β u β , ρ β (( v β ) − ( u β ) ) , ρ β u β v β , ρ β | v β | + p β , (cid:18) ρ β | v β | + 2 p β (cid:19) v β , (cid:18) ρ β | v β | + 2 p β (cid:19) u β (cid:19) . Note that | v β | = | ( v ′ ) β + δ p e | = ε + ˜ ε + o ( δ ) on Ω . The matrix (3.1) on the considered set is of the form ρ − ρ K ρ v β ρ u β ρ v β ρ | v β | + p − p + ρ u β v β ρ u β ρ u β v β ρ | u β | + p − p +12 ρ | v β | + p − p + (cid:0) ρ | v β | + 2 p (cid:1) v β (cid:0) ρ | v β | + 2 p (cid:1) u β . The determinant of the submatrix consisting of the first, second, and third rows is( p − p + ) (cid:0) ( ρ − ρ K )( p − p + ) − ρ ρ K | v β | (cid:1) . So the corresponding matrix Z A is of rank 3 once we know that p = p + and | v β | = ( ρ − ρ K )( p − p + ) ρ ρ K . (3.3)We have p ≈ p − (cid:16) ρ K ρ − (cid:17) = p + once we know that p + > p − , and so it remains to verify (3.3).Since all above mentioned quantities are bounded and ρ − ρ K is negligible, the right hand side of (3.3) canbe made arbitrarily close to zero. We need to show that | v β | is far away of zero. According to [16, Sections3.2 & 3.8] we have | v β | ≈ ε ( v K , ρ K , ρ K , p − , p + ) + ˜ ε ( v K , ρ K , ρ K , p − , p + ) = 4 ( p + − p − )( p + + p − ) ρ − (3 p + + p − )(3 p − + p + ) . Consequently, (3.3) is fulfilled and it is allowed to consider also the intended small perturbations. The Youngmeasure ν t,x = 12 δ ( ρ α ,ρ α v α , ρ α | v α | + p α ) + 12 δ ( ρ β ,ρ β v β , ρ β | v β | + p β ) is a measure value solution which, due to Theorem 3.1, cannot be generated by weak solutions.We get the following claim as a result of the previous considerations. Theorem 3.2.
There exists a measure-valued solution to (2.1) with non-constant entropy, emanating fromcertain Riemann initial data, which cannot be generated by a sequence of weak solutions.
Acknowledgement:
The research of V.M. was supported by the Czech Science Foundation, Grant Agree-ment GA18–05974S, in the framework of RVO:67985840.
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