INSTITUTE of MATHEMATICS. ACADEMY of SCIENCES of the CZECH REPUBLIC. On the weak solutions to the equations of a compressible heat conducting gas

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1 INSTITUTE of MATHEMATICS Academy of Sciences Czech Republic INSTITUTE of MATHEMATICS ACADEMY of SCIENCES of the CZECH REPUBLIC On the weak solutions to the equations of a compressible heat conducting gas Elisabetta Chiodaroli Eduard Feireisl Ondřej Kreml Preprint No PRAHA 03

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3 On the weak solutions to the equations of a compressible heat conducting gas Elisabetta Chiodaroli Eduard Feireisl Ondřej Kreml Mathematisches Institut, Universität Leipzig Augustusplatz 0, D Leipzig, Germany Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 5, 5 67 Praha, Czech Republic Institut für Mathematik, Universität Zürich Winterthurerstrasse 90, CH-8057 Zürich, Switzerland Abstract We consider the weak solutions to the Euler-Fourier system describing the motion of a compressible heat conducting gas. Employing the method of convex integration, we show that the problem admits infinitely many global-in-time weak solutions for any choice of smooth initial data. We also show that for any initial distribution of the density and temperature, there exists an initial velocity such that the associated initial-value problem possesses infinitely many solutions that conserve the total energy. Introduction The concept of weak solution has been introduced in the mathematical theory of systems of nonlinear) hyperbolic conservation laws to incorporate the inevitable singularities in their solutions that may develop in a finite time no matter how smooth and small the data are. As is well known, however, many nonlinear problems are not well posed in the weak framework and several classes of The research of E.F. leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme FP7/007-03)/ ERC Grant Agreement The work of O.K. is part of the SCIEX project.5

4 admissible weak solutions have been identified to handle this issue. The implications of the Second law of thermodynamics have been widely used in the form of various entropy conditions in order to identify the physically relevant solutions. Although this approach has been partially successful when dealing with systems in the simplified D geometry, see Bianchini and Bressan [4], Bressan [5], Dafermos [7], Liu [4], among others, the more realistic problems in higher spatial dimensions seem to be out of reach of the theory mostly because the class of entropies is rather poor consisting typically of a single physical) entropy. Recently, De Lellis and Székelyhidi [9] developed the method of convex integration cf. Müller and Šverák [5]) to identify a large class of weak solutions to the Euler system violating the principle of well-posedness in various directions. Besides the apparently non-physical solutions producing the kinetic energy cf. Shnirelman [8]), a large class of data has been identified admitting infinitely many weak solutions that comply with a major part of the known admissibility criteria, see De Lellis and Székelyhidi [8]. In this paper, we develop the technique of [8] to examine the well-posedness of the full Euler- Fourier system: t ϱ + div x ϱu) = 0,.) t ϱu) + div x ϱu u) + x pϱ, ϑ) = 0,.) t ϱeϱ, ϑ)) + div x ϱeϱ, ϑ)u) + div x q = pϱ, ϑ)div x u,.3) where ϱt, x) is the mass density, u = ut, x) the velocity field, and ϑ the absolute) temperature of a compressible, heat conducting gas, see Wilcox [9]. For the sake of simplicity, we restrict ourselves to the case of perfect monoatomic gas, for which the pressure pϱ, ϑ) and the specific internal energy eϱ, ϑ) are interrelated through the constitutive equations: pϱ, ϑ) = ϱeϱ, ϑ), pϱ, ϑ) = aϱϑ, a > 0..4) 3 Although the system. -.3) describes the motion in the absence of viscous forces, we suppose that the fluid is heat conductive, with the heat flux q determined by the standard Fourier law: The problem. -.3) is supplemented with the initial data q = κ x ϑ, κ > 0..5) ϱ0, ) = ϱ 0, ϱu)0, ) = ϱ 0 u 0, ϑ0, ) = ϑ 0 in..6) In addition, to avoid the effect of the kinematic boundary, we consider the periodic boundary conditions, meaning the physical domain will be taken the flat torus = T 3 = [0, ] {0,} ) 3.

5 The first part of the paper exploits the constructive aspect of convex integration. We present a variable coefficients variant of a result of De Lellis-Székelyhidi [8] and show the existence of infinitely many global-in-time weak solutions to the problem. -.6) for any physically relevant choice of smooth) initial data. Here, physically relevant means that the initial distribution of the density ϱ 0 and the temperature ϑ 0 are strictly positive in. These solutions satisfy also the associated entropy equation; whence they comply with the Second law of thermodynamics. Similarly to their counterparts constructed in [8], these wild weak solutions violate the First law of thermodynamics, specifically, the total energy at any positive time is strictly larger than for the initial data. In order to eliminate the non-physical solutions, we therefore impose the total energy conservation in the form: ) ) Et) = ϱ u + eϱ, ϑ) t, ) dx = ϱ 0 u 0 + eϱ 0, ϑ 0 ) dx = E 0 for a.a.) t 0, T )..7) Following [] we show that the system. -.3), augmented with the total energy balance.7), satisfies the principle of weak-strong uniqueness. Specifically, the weak and strong solutions emanating from the same initial data necessarily coincide as long as the latter exists. In other words, the strong solutions are unique in the class of weak solutions. This property remains valid even if we replace the internal energy equation.3) by the entropy inequality t ϱsϱ, ϑ))+div x ϱsϱ, ϑ)u)+div x q ϑ ) q xϑ, ϑdsϱ, ϑ) Deϱ, ϑ)+pϱ, ϑ)d ϑ ),.8) ϱ in the spirit of the theory developed in []. Although the stipulation of.7) obviously eliminates the non-physical energy producing solutions, we will show that for any initial data ϱ 0, ϑ 0 there exists an initial velocity u 0 such that the problem. -.6) admits infinitely many global-in-time weak solutions that satisfy the total energy balance.7). The paper is organized as follows. After a brief introduction of the concept of weak solutions in Section, we discuss the problem of existence of infinitely many solutions for arbitrary initial data, see Section 3. In particular, we prove a variable coefficients variant of a result of De Lellis and Székelyhidi [8] and employ the arguments based on Baire s category. In Section 4, we show the weak-strong uniqueness principle for the augmented system and then identify the initial data for which the associated solutions conserve the total energy. The paper is concluded by some remarks on possible extensions in Section 5. By global-in-time solutions we mean here solutions defined on [0, T ) for any T > 0. For discussion about solutions defined on [0, ) see Section 5. 3

6 Weak solutions To simplify presentation, we may assume, without loss of generality, that a = κ =. We say that a trio [ϱ, ϑ, u] is a weak solution of the problem. -.6) in the space-time cylinder 0, T ) if: the density ϱ and the temperature ϑ are positive in 0, T ) ; T for any test function ϕ C c 0 ϱ t ϕ + ϱu x ϕ) dx dt = ϱ 0 ϕ0, ) dx.) [0, T ) ); T 0 ϱu t ϕ + ϱu u : x ϕ + ϱϑdiv x ϕ) dx dt = ϱ 0 u 0 ϕ0, ) dx.) for any test function ϕ C c [0, T ) ; R 3 ); T 3 [ϱϑ tϕ + ϱϑu x ϕ] x ϑ x ϕ ϱϑdiv x uϕ) 0 dx dt = ϱ 0 ϑ 0 ϕ0, ) dx.3) for any test function ϕ C c [0, T ) ). As a matter of fact, the weak solutions we construct in this paper will be rather regular with the only exception of the velocity field. In particular, the functions ϱ, ϑ, and even div x u will be continuously differentiable in [0, T ], and, in addition, ϑ L 0, T ; W,p )), t ϑ L p 0, T ; L p )) for any p <. Thus the equations.),.3) will be in fact satisfied pointwise a.a. in 0, T ). As for the velocity field, we have u C weak [0, T ]; L ; R 3 )) L 0, T ) ; R 3 ), div x u C[0, T ] R 3 ). 4

7 3 Second law is not enough Our first objective is to show the existence of infinitely many solutions to the Euler-Fourier system for arbitrary smooth) initial data. Theorem 3. Let T > 0. Let the initial data satisfy ϱ 0 C 3 ), ϑ 0 C ), u 0 C 3 ; R 3 ), ϱ 0 x) > ϱ > 0, ϑ 0 x) > ϑ > 0 for any x. 3.) Then the initial-value problem. -.6) admits infinitely many weak solutions in 0, T ) belonging to the class: ϱ C [0, T ] ), t ϑ L p 0, T ; L p )), xϑ L p 0, T ; L p ; R 3 3 )) for any p <, u C weak [0, T ]; L ; R 3 )) L 0, T ) ; R 3 ), div x u C [0, T ] ). Remark 3. Using the maximal regularity theory for parabolic equations see Amann [3], Krylov [3]) we observe that ϑ is a continuous function of the time variable t ranging in the interpolation space [L p ); W,p )] α for any p < finite and any α 0, ). Thus it is possible to show that the conclusion of Theorem 3. remains valid if we assume that ϑ 0 [L p ); W,p )] α for sufficiently large p < and 0 < α <, ϑ 0 > 0 in, where [, ] α denotes the real interpolation. In particular, the solution ϑt, ) will remain in the same regularity class for any t [0, T ]. The rest of this section is devoted to the proof of Theorem Reformulation Following Chiodaroli [6], we reformulate the problem in the new variables ϱ, ϑ, and w = ϱu obtaining, formally t ϱ + div x w = 0, 3.) t w + div x w w ϱ 5 ) + x ϱϑ) = 0, 3.3)

8 with 3 ϱ tϑ + w x ϑ) = ϑ ϑdiv x w + ϑ xϱ w. 3.4) ϱ Next, we take the following ansatz for the density: ϱt, x) = ϱ 0 x) ht)div x ϱ 0 x)u 0 x)) t, x), h C [0, T ], h0) = 0, h 0) =, ϱ 0 x) ht)div x ϱ 0 x)u 0 x)) > ϱ for all t [0, T ], x. 3.5) Accordingly, we write w in the form of its Helmholtz decomposition w = v + x Ψ, div x v = 0, Ψ = h t)div x ϱ 0 u 0 ), Ψ dx = 0. Obviously, by virtue of the hypotheses 3.) imposed on the initial data, we have C [0, T ] ), x Ψ C [0, T ] ; R 3 ), 0, ) = ϱ 0, w 0 = ϱ 0 u 0. Moreover, the equation of continuity 3.) is satisfied pointwise in 0, T ), while the remaining two equations 3.), 3.3) read ) v + x Ψ) v + x Ψ) t v + div x + x ϑ + t Ψ) = 0, div x v = 0, 3.6) v0, ) = v 0 = ϱ 0 u 0 x div x ϱ 0 u 0 ), 3.7) 3 t ϑ + v + x Ψ) x ϑ ) = ϑ ϑ Ψ + ϑ x v + xψ), ϑ0, ) = ϑ ) 3. Internal energy and entropy equations For a given vector field v L 0, T ) ; R 3 ), the internal energy equation 3.8) is linear with respect to ϑ and as such admits a unique solution ϑ = ϑ[v] satisfying the initial condition ϑ0, ) = ϑ 0. Moreover, the standard L p theory for parabolic equations see e.g. Krylov [3]) yields ϑt, x) > 0 for all t [0, T ], x, t ϑ L p 0, T ; L p )), xϑ L p 0, T ; L p ; R 3 3 ) for any p <, 3.9) where the bounds depend only on the data and v L 0,T ) ;R 3 ). Dividing 3.8) by ϑ we deduce the entropy equation ) ) ϑ 3/ ϑ 3/ t log + v + x Ψ) x log = logϑ) + x logϑ), 3.0) where we have used the identity Ψ = t. We note that, given the regularity of the solutions in Theorem 3., the entropy equation 3.0) and the internal energy equation 3.8) are equivalent. In particular, the weak solutions we construct are compatible with the Second law of thermodynamics. 6

9 3.. Uniform bounds Introducing a new variable we may rewrite 3.0) as Z = log ) ϑ 3/ t Z + v + x Ψ 8 9 x log) ) x Z = 3 Z xz + 3 log) x log). 3.) Applying the standard parabolic comparison principle to 3.) we conclude that Z is bounded only in terms of the initial data and the time T. Consequently, the constants ϑ, ϑ can be taken in such a way that 0 < ϑ ϑ[v]t, x) ϑ for all t [0, T ], x. 3.) We emphasize that the constants ϑ, ϑ are independent of v - a crucial fact that will be used in the future analysis. 3.3 Reduction to a modified Euler system Summing up the previous discussion, our task reduces to finding infinitely many) solutions to the problem ) v + x Ψ) v + x Ψ) t v + div x + x ϑ[v] + t Ψ 3 ) χ = 0, div x v = 0, v0, ) = v 0, 3.3) with a suitable spatially homogeneous function χ = χt). Following the strategy and notation) of De Lellis and Székelyhidi [8], we introduce the linear system t v + div x U = 0, div x v = 0, v0, ) = v 0, vt, ) = v T, 3.4) together with the function e, e[v] = χ 3 ϑ[v] 3 tψ, 3.5) with a positive function χ C[0, T ] determined below. Furthermore, we introduce the space Rsym,0 3 3 of symmetric traceless matrices, with the operator norm λ max [U] the maximal eigenvalue of U Rsym,

10 Finally, we define the set of subsolutions { X 0 = v v L 0, T ) ; R 3 ) C 0, T ) ; R 3 ) C weak [0, T ]; L ; R 3 )), 3.6) inf t ε,t ),x v satisfies 3.4) with some U C 0, T ) ; Rsym,0), 3 3 { e[v] 3 [ ]} } v + λ x Ψ) v + x Ψ) max U > 0 for any 0 < ε < T. Note that X 0 is different from its analogue introduced by Chiodaroli [6] and De Lellis and Székelyhidi [8], in particular, the function e[v] depends on the field v. As shown by De Lellis and Székelyhidi [8], we have the pointwise) inequality where the identity holds only if w 3 λ max [w w U], w R 3, U R 3 3 sym,0, U = w w 3 w I. Consequently, by virtue of 3.), there exists a constant c depending only on the initial data [ϱ 0, ϑ 0, u 0 ] such that sup t [0,T ] vt, ) L ;R 3 ) < c for all v X ) Next, we choose the function χ C[0, T ] in 3.5) so large that [ ] 3 λ v0 + x Ψ) v 0 + x Ψ) max < χ 3 ϑ[v 0] 3 tψ e[v 0 ] for all t, x) [0, T ], in particular, the function v 0 = v 0 x), together with the associated tensor U 0, belongs to the set X 0, where v 0 = v T. We define a topological space X as a completion of X 0 in C weak [0, T ]; L ; R 3 )) with respect to the metric d induced by the weak topology of the Hilbert space L ; R 3 ). As we have just observed, the space X 0 is non-empty as v = v 0 is in X 0. Finally, we consider a family of functionals T v + x Ψ ) I ε [v] = e[v] dx dt for v X, 0 < ε < T. 3.8) ε As a direct consequence of the parabolic regularity estimates 3.9), we observe that e[v] e[w] in C[0, T ] ) whenever v w in X; 3.9) 8

11 therefore each I ε is a compact perturbation of a convex functional; whence lower semi-continuous in X. In order to proceed, we need the following crucial result that may be viewed as a variable coefficients counterpart of [8, Proposition 3]. Proposition 3. Let v X 0 such that I ε [v] < α < 0, 0 < ε < T/. There there is β = βα) > 0 and a sequence {v n } n= X 0 such that v n v in C weak [0, T ]; L ; R 3 )), lim inf n I ε[v n ] I ε [v] + β. We point out that the quantity β = βα) is independent of ε and v. Postponing the proof of Proposition 3. to the next section, we complete the proof of Theorem 3. following the line of arguments of [8]. To begin, we observe that cardinality of the space X 0 is infinite. Secondly, since each I ε is a bounded lower semi-continuous functional on a complete metric space, the points of continuity of I ε form a residual set in X. The set C = m> { v X I/m [v] is continuous }, being an intersection of a countable family of residual sets, is residual, in particular of infinite cardinality, see De Lellis and Székelyhidi [8] for a more detailed explanation of these arguments. Finally, we claim that for each v C we have whence v + x Ψ I /m [v] = 0 for all m > ; = e[v] χ 3 ϑ[v] 3 tψ, U = v + xψ) v + x Ψ) v + x Ψ I for a.a. t, x) 0, T ), 3 in other words, the function v is a weak solution to the problem 3.3). Indeed, assuming I /m [v] < α < 0, we first find a sequence {u n } n= X 0 such that u n v in C weak [0, T ]; L ; R 3 )), I /m [u n ] < α. Then for each u n we use Proposition 3. and together with standard diagonal argument we obtain a sequence {v n } n= X 0 such that v n v in C weak [0, T ]; L ; R 3 )), lim inf n I /m[v n ] I /m [v] + β, β > 0, in contrast with the fact that v is a point of continuity of I /m. 9

12 3.4 Proof of Proposition 3. The proof of Proposition 3. is based on a localization argument, where variable coefficients are replaced by constants. The fundamental building block is the following result proved by De Lellis and Székelyhidi [8, Proposition 3], Chiodaroli [6, Section 6, formula 6.9)]: Lemma 3. Let [T, T ], T < T, be a time interval and B R 3 a domain. Let r 0, ), Ṽ R3 be constant fields such that 0 < r < r < r, Ṽ < V. Suppose that satisfies the linear system v C weak [T, T ]; L B, R 3 )) C T, T ) B; R 3 ) t v + div x U = 0, div x v = 0 in T, T ) B with the associated field U C T, T ) B; Rsym,0) 3 3 such that [ 3 v + λ Ṽ) v + Ṽ) ] max U < e in T, T ) B r for a certain function e C[T ; T ] B). Then there exist sequences {w n } n= Cc T, T ) B; R 3 ), {Y n } n= Cc T, T ) B; Rsym,0) 3 3 such that v n = v + w n, U n = U + Y n satisfy and lim inf n T T B t v n + div x U n = 0, div x v n = 0 in T, T ) B [ 3 λ vn + Ṽ) v n + Ṽ) ] max U n < e in T, T ) B, r v n v C weak [T, T ]; L B; R 3 )), v n v dx dt Λ r, r, V, e L T,T ) B) Remark 3. Note that Ṽ is constant in Lemma 3.; whence t v = t v + Ṽ). ) T T B e ) v + Ṽ dx dt. r 3.0) Remark 3.3 It is important that the constant Λ depends only on the quantities indicated explicitly in 3.0), in particular Λ is independent of v, of the length of the time interval, and of the domain B. 0

13 3.4. Localization principle The scale invariance encoded in 3.0) can be used for showing a variable coefficients variant of Lemma 3., specifically when both r and Ṽ are sufficiently smooth functions of t and x. Lemma 3. Let C [T, T ] ), V C [T, T ] ; R 3 ), T < T be functions satisfying Suppose that solves the linear system 0 < r < t, x) < r, Vt, x) < V for all t, x. v C weak [T, T ]; L, R 3 )) C T, T ) ; R 3 ) t v + div x U = 0, div x v = 0 in T, T ) with the associated field U C T, T ) B; Rsym,0) 3 3 such that [ ] 3 v + V) v + V) λ max U < e δ in T, T ) 3.) for some e C[T ; T ] B) and δ > 0. Then there exist sequences {w n } n= Cc T, T ) ; R 3 ), {Y n } n= Cc T, T ) ; Rsym,0) 3 3 such that v n = v + w n, U n = U + Y n satisfy and lim inf n T T t v n + div x U n = 0, div x v n = 0 in T, T ) 3.) [ ] 3 λ vn + V) v n + V) max U n < e in CT ; T ) ), 3.3) v n v C weak [T, T ]; L ; R 3 )), 3.4) v n v dx dt Λ r, r, V, e L T,T ) ) ) T T e v + V ) dx dt. 3.5) Remark 3.4 The role of the positive parameter δ in 3.) is only to say that the inequality 3.) is strict, otherwise the conclusion of the lemma is independent of the specific value of δ. Remark 3.5 In view of 3.4), the convergence formula 3.5) may be equivalently replaced by T T v + V lim inf n T T v n + V dx dt + Λ r, r, V, e L T,T ) ) dx dt 3.6) ) T T e v + V ) dx dt.

14 Proof: We start with an easy observation that there exists ε = ε δ, e ) such that 3 [ v+v) v+v) λ max ] [ ] U v+ṽ) v+ṽ) λmax U < δ, r 4 v+v v+ṽ r < δ 4 3.7) whenever [ ] 3 v + V) v + V) λ max U < e, r < ε, V Ṽ < ε. For δ appearing in 3.), we fix ε = ε δ, e L T,T ) )) as in 3.7) and find a finite) decomposition of the set T, T ) such that [T, T ] = N i=q i, Q i = T i, T i ) B i, Q i Q j = for i j, sup Q i inf Q i < ε, sup Q i V V dx dt Q i Q i < ε, where the number N depends on ε and the Lipschitz constants of, V in [T, T ]. Now, we apply Lemma 3. on each set Q i with the choice of parameters In accordance with 3.), 3.7), we have r = sup, Ṽ = V dx dt. Q i Q i Q i [ 3 v + λ Ṽ) v + Ṽ) ] max U < e δ r in Q i. Under these circumstances, Lemma 3. yields a sequence of smooth functions v i n, U i n, with v v i n, U U i n compactly supported in Q i, such that t v i n + div x U i n = 0, div x v i n = 0 in Q i, [ 3 v i λ n + Ṽ) vi n + Ṽ) ] max U i n < e δ r, 3.8) v i n v in C weak [T i, T i ], L B i )),

15 and lim inf n Q i vn i v dx dt Λ r, r, V, e L T,T ) ) ) Q i e v + Ṽ r δ ) dx dt. 3.9) In view of 3.7), we replace r by and Ṽ by V in 3.8) to obtain [ 3 v i λ n + V) v i ] n + V) max U n < e in Q i. As v n, U n are compactly supported perturbations of v, U in Q i, we may define v n t, x) = v i nt, x), U n = U i n for any t, x) Q i, i =,..., N. In accordance with the previous discussion, v n, U n satisfy ). In order to see 3.5), use 3.7) to observe that e v + Ṽ r δ ) > e v + V 3δ ) > 0 in Q i ; 4 whence, making use of the hypothesis 3.), specifically of the fact that e v + V > δ, we may infer that e v + V 3δ ) e 4 4 Thus, summing up the integrals in 3.9) we get 3.5). v + V ) in Q i. Q.E.D Application to functionals I ε Fixing ε 0, T/) we complete the proof of Proposition 3.. Given e C[0, T ] ), we introduce the spaces { X 0,e = v v C 0, T ) ; R 3 ) C weak [0, T ]; L ; R 3 )) 3.30) v satisfies 3.4) with some U C 0, T ) ; R 3 3 sym,0), 3

16 [ ] } 3 v + λ x Ψ) v + x Ψ) max U < e for t 0, T ), x, along with the associated functionals I ε,e [v] = T ε v + x Ψ ) e The following assertion is a direct consequence of Lemma 3.. Lemma 3.3 Let v X 0,e, e C[0, T ] ), 0 < ε < T/ be such that I ε,e [v] < α < 0. dx dt for v X, 0 < ε < T/. 3.3) There there is β = βα, e L 0,T ) )) > 0, independent of ε, and a sequence {v n } n>0 X 0,e such that v n v in [0, ε], v n v in C weak [0, T ]; L ; R 3 )), lim inf n I ε,e[v n ] I ε,e [v] + β. Remark 3.6 We have used Lemma 3. with 3.6), where, by virtue of Jensen s inequality, T ε e v + V ) dx dt T e T ε) ε v + V ) dx dt) α T ε). Finally, we show how Lemma 3.3 implies Proposition 3.. Under the hypotheses of Proposition 3. and in accordance with the definition of the space X 0, we find δ > 0 and a function e C[0, T ] ) such that e e[v], e e[v] δ whenever t [ε, T ], and Thus, we have I ε,e [v] = T ε v X 0,e. v + x Ψ ) e[v] + δ dx dt = I ε [v] + T ε) δ < α/ < 0 as soon as δ > 0 was chosen small enough. Consequently, by virtue of Lemma 3.3, there is a sequence of functions {v n } n= and β = βα) > 0 such that v n X 0,e, v n v in [0, ε], 4

17 and v n v in C weak [0, T ]; L ; R 3 )), lim inf n Moreover, in accordance with 3.9), I ε,e [v n ] I ε [v n ] = T ε I ε,e[v n ] I ε,e [v] + β = I ε [v] + β + T ε) δ. e[v n ] e[v] + δ dx T ε) δ as n ; whence we may infer that lim inf I ε[v n ] I ε [v] + β. n Finally, it remains to observe that v n X 0 for all n large enough. To this end, note that 3 λ max [ ] vn + x Ψ) v n + x Ψ) U = 3 [ ] v + λ x Ψ) v + x Ψ) max U < e e[v] = e[v n ] for all t [0, ε], while [ ] 3 λ vn + x Ψ) v n + x Ψ) max U < e = e[v] δ e[v n ] δ/ for all t [ε, T ] for all n large enough. We have proved Proposition Dissipative solutions The solutions of the Euler-Fourier system constructed in Section 3 suffer an essential deficiency, namely they do not comply with the First law of thermodynamics, meaning, they violate the total energy conservation.7). On the other hand, the initial data in 3.) are smooth enough for the problem to possess a standard classical solution existing on a possibly short time interval 0, T max ), see e.g. Alazard [], []. Note that the Euler-Fourier system fits also in the general framework and the corresponding existence theory developed by Serre [6], [7]. As the classical solutions are unique in their regularity class) and obviously satisfy the total energy balance.7), the latter can be added to. -.3) as an admissibility condition. The weak solutions of. -.6) satisfying.7) will be called dissipative solutions. 4. Relative entropy energy) and weak-strong uniqueness Following [] we introduce the relative entropy functional E ϱ, ϑ, u r, Θ, U ) 4.) 5

18 = ϱ u U + H Θ ϱ, ϑ) H ) Θr, Θ) ϱ r) H Θ r, Θ) ϱ where H Θ is the ballistic free energy, H Θ ϱ, ϑ) = ϱ eϱ, ϑ) Θsϱ, ϑ) ) = ϱ 3 ϑ Θ log )) ϑ 3/. ϱ Repeating step by step the arguments of [] we can show that any dissipative solution of the problem. -.6) satisfies the relative entropy inequality: τ 0 τ [ E ϱ, ϑ, u r, Θ, U )] t=τ t=0 + τ 0 dx, Θ xϑ ϑ ) dx dt 4.) ϱu u) t U + ϱu u) u : x U pϱ, ϑ)div x U ) dx dt ϱ sϱ, ϑ) sr, Θ) ) t Θ + ϱ sϱ, ϑ) sr, Θ) ) u x Θ ) dx dt 0 τ + ϱ ) t pr, Θ) ϱ ) 0 r r u xpr, Θ) for any trio of smooth test functions dx + r, Θ, U, r > 0, Θ > 0. We report the following result [0, Theorem 6.]. τ 0 x ϑ ϑ xθ dx dt Theorem 4. [Weak-strong uniqueness] Let [ϱ, ϑ, u] be a dissipative weak) solution of the problem. -.6), emanating from the initial data [ϱ 0, ϑ 0, u 0 ] satisfying 3.), such that 0 < ϱ < ϱt, x) < ϱ, 0 < ϑ < ϑt, x) < ϑ, ut, x) < u for a.a. t, x) 0, T ). Suppose that the same problem with the same initial data) admits a classical solution [, ϑ, ũ] in 0, T ). Then ϱ, ϑ ϑ, u ũ. 6

19 Remark 4. Here, classical means that all the necessary derivatives appearing in the equations are continuous functions in [0, T ]. Remark 4. The proof of Theorem 4. is based on taking r =, Θ = ϑ, U = ũ as test functions in the relative entropy inequality 4.) and making use of a Gronwall type argument. This has been done in detail in [0, Section 6] in the case of a viscous fluid satisfying the Navier-Stokes-Fourier system. However, the same arguments can be used to handle the inviscid case provided the solutions are uniformly bounded on the existence interval. Remark 4.3 As the proof of Theorem 4. is based on the relative entropy inequality 4.), the conclusion remains valid if we replace the internal energy equation.3) by the entropy inequality.8) as long as we require.7). 4. Infinitely many dissipative solutions Apparently, the stipulation of the total energy balance.7) eliminates the non-physical solutions obtained in Theorem 3., at least in the case of regular initial data. As we will see, the situation changes if we consider non-smooth initial data, in particular the initial velocity field u 0 belonging only to L ; R 3 ). Our final goal is the following result. Theorem 4. Let T > 0 be given. Let the initial data ϱ 0, ϑ 0 be given, satisfying ϱ 0, ϑ 0 C ), ϱ 0 x) > ϱ > 0, ϑ 0 x) > ϑ > 0 for any x. 4.3) Then there exists a velocity field u 0, u 0 L ; R 3 ), such that the problem. -.6) admits infinitely many dissipative weak) solutions in 0, T ), with the initial data [ϱ 0, ϑ 0, u 0 ]. Remark 4.4 As we shall see below, the solutions obtained in the proof of Theorem 4. enjoy the same regularity as those in Theorem 3., in particular, the equation of continuity.) as well as the internal energy balance.3) are satisfied pointwise a.a.) in 0, T ). 7

20 Remark 4.5 In general, the initial velocity u 0 depends on the length of the existence interval T. See Section 5 for more discussion concerning possible extension of the solutions to [0, ). The remaining part of this section is devoted to the proof of Theorem 4. that may be viewed as an extension of the results of Chiodaroli [6] and De Lellis-Székelyhidi [8] to the case of a heat conducting fluid. 4.. Suitable initial data Following the strategy of [8] our goal is to identify suitable initial data u 0 for which the associated weak) solutions of the momentum equation dissipate the kinetic energy. In contrast with [8], however, we have to find the initial data for which the kinetic energy decays sufficiently fast in order to compensate the associated production of heat. The velocity field v = ϱu we look for will be solenoidal, in particular, we focus on the initial data satisfying div x ϱ 0 u 0 ) = 0. This assumption simplifies considerably the ansatz introduced in Section 3., specifically, whence the problem reduces to solving ϱ = = ϱ 0 x), v = ϱu, div x v = 0, Ψ 0; ) v v t v + div x + x ϑ[v] 3 ) χ = 0, div x v = 0, v0, ) = v 0, div x v 0 = 0, 4.4) for a suitable spatially homogeneous function χ = χt). Mimicking the steps of Section 3.3 we introduce the quantity As the anticipated solutions satisfy the energy of the system reads Et) = v e[v] = χ 3 ϑ[v]. 4.5) v + 3 ϑ[v] = e[v], ) t, ) dx = χt) = χt). 4.6) 8

21 Consequently, in accordance with the construction procedure used in Section 3.3, it is enough to find a suitable constant χ and the initial velocity field v 0 such that div x v 0 = 0, E 0 = v ϑ 0 ) dx = χ, and the associated space of subsolutions X 0 defined in 3.6) with x Ψ = 0) is non-empty. This is the objective of the remaining part of this section. 4.. Dissipative data for the Euler system Similarly to 3.30), we introduce the set of subsolutions X 0,e [T, T ] = { v v C T, T ) ; R 3 ) C weak [T, T ]; L ; R 3 )) 4.7) v satisfies 3.4) with some U C T, T ) ; R 3 3 [ ] } 3 v v λ max U < e for t T, T ), x, where e C[T, T ] ). The following result may be seen as an extension of [8, Proposition 5]: sym,0), Lemma 4. Suppose that v v 0 x), together with the associated field U v 0, belong to the set of subsolutions X 0,e [0, T ]. Then for any τ 0, T ) and any ε > 0, there exist τ 0, T ), τ τ < ε and w X 0,e [τ, T ], such that wτ, ) = eτ, ), 4.8) Remark 4.6 Note that, thanks to 4.8), w v, U w 0 in a left) neighborhood of T. wt, ) wτ, ) strongly) in L ; R 3 ) as t τ +. Remark 4.7 The result is probably not optimal; one should be able, with greater effort, to show the same conclusion with τ = τ. 9

22 Proof: We construct the function w as a limit of a sequence {w k } k= X 0,e [0, T ], w k w in C weak [0, T ]; L ; R 3 )), where w k will be obtained recursively, with the starting point w 0 = v v 0, τ 0 = τ, ε 0 = ε. More specifically, we construct the functions w k, together with τ k, ε k, k =,... satisfying: w k X 0,e [0, T ], supp[w k w k ] τ k ε k, τ k + ε k ), where 0 < ε k < ε k ; 4.9) dw k, w k ) < k, sup t 0,T ) w k w k ) w m dx < for all m = 0,..., k, 4.0) k recalling that d is the metric induced by the weak topology of the Hilbert space L ; R 3 ). there exists τ k, such that where w k τ k τ k ε k, τ k + ε k ) w k τ k, ) dx α k = w k t, ) dx + λ α ε k 4.) k τ k, ) dx + λ α ε k for all t τ k ε k, τ k + ε k ), k τk +ε k τ k ε k and λ > 0 is constant independent of k. e w k ) dx dt > 0, Supposing we have already constructed w 0,..., w k we find w k enjoying the properties ). To this end, we first compute α k = τk +ε k τ k ε k e w k ) 0 dx dt for a certain 0 < ε k < ε k

23 and observe that α k = τk +ε k ε k ε k τ k ε k e w k ) dx dt e w k ) τ k ) dx > 0 for ε k 0 as w k is smooth in 0, T ). Consequently, by the same token, we can choose ε k > 0 so small that w k ε k τk +ε k τ k ε k w k w k dx dt + Λ, e L 0,T ) )) α 4ε k 4.) k t, ) dx + Λ, e L 0,T ) )) α 8ε k k τ k, ) dx + Λ, e L 0,T ) )) α 6ε k for all t τ k ε k, τ k + ε k ), k where Λ, e L 0,T ) )) > 0 is the universal constant from Lemma 3.. Applying Lemma 3. in the form specified in Remark 3.5 we obtain a function w k X 0,e such that supp[w k w k ] τ k ε k, τ k + ε k ), dw k, w k ) <, sup k w k w k ) w m dx <, m = 0,..., k 4.3) k and τk +ε k τ k ε k w k t 0,T ) dx dt τk +ε k τ k ε k w k dx dt + Λ, e L 0,T ) )) ε k α k, 4.4) where we have applied Jensen s inequality to the last integral in 3.6). Finally, the relations 4.), 4.4) yield 4.) with some τ k τ k ε k, τ k + ε k ), λ = Λ/6. Now, by virtue of 4.0), there is w such that Moreover, 4.9) implies i) ii) for any δ > 0 there is k = k 0 δ) such that w k w in C weak [0, T ]; L ; R 3 )). 4.5) τ k τ 0, T ), τ τ < ε; wt, ) = w k t, ) = w k0 t, ) for all t 0, τ δ) τ + δ, T ), k k )

24 In particular, 4.6) yields Next, in view of 4.), therefore whence, finally, w X 0,e [τ, T ], and w v, U w 0 in a left) neighborhood of T. w k t, ) dx Y uniformly for t τ k ε k, τ k + ε k ), 4.7) α k ε k = ε k τk +ε k τ k ε k Combining 4.9) with 4.0), 4.5) we get which implies 4.8). Indeed we have = e w k ) dx dt 0; 4.8) w k τ, ) dx eτ, ) dx. 4.9) w k τ, ) wτ, ) in L ; R 3 ) w n w m τ, ) dx w n τ, ) dx w m τ, ) dx w n w m ) w m τ, ) dx for all n > m, where, by virtue of 4.0), w n w m ) w m τ, ) dx = n m k=0 w k+ w k ) w m τ, ) dx 0 for m. Q.E.D.

25 4..3 Construction of suitable initial data for the Euler-Fourier system Fixing ϱ 0, ϑ 0 satisfying 4.3) and ϱ = ϱ 0 we can use 3.) to deduce that there is a constant ϑ depending only on [ϱ 0, ϑ 0 ] such that ϑ[v] ϑ, whence 3 p, ϑ[v]) < P on the whole interval [0, T ], 4.0) with P independent of v. Next, we estimate the difference ϑ ϑ 0 satisfying the equation t ϑ ϑ 0 ) + v x ϑ ϑ 0 ) 3 ϑ ϑ 0) = v x ϑ ϑ ϑv x. Consequently, using 4.0) and the comparison principle, we deduce that ϑ[v]t, ) ϑ 0 c + v L 0,T ) ;R 3 )) t for all t [0, T ]. 4.) We take v 0 C ), div x v 0 = 0, and a constant χ 0 in such a way that ) 3 λ v0 v 0 max < χ 0 3 ϱ 0ϑ 0. 4.) Moreover, for any χ > χ 0, K > 0 given, it is easy to construct a function χ C[0, T ] such that χ0) = χt ) = χ 0, χt) > χ 0 for all t 0, T ), there is τ 0, T ) and ε > 0 such that max χt) = χ; t 0,T ) χτ) χ 0 > χ, χt) < χτ) Kt τ) for all t τ, T ) whenever τ τ < ε. 4.3) Consequently, we have provided v v 0 X 0,e [0, T ] with U 0) et, x) = χt) 3 ϱ 0ϑ 0. Applying Lemma 4., we find a function w X 0,e [τ, T ], with the corresponding field U w, such that wτ, ) = χτ) 3 ϱ 0ϑ 0 > χ + χ 0 3 ϱ 0ϑ 0, 3

26 and w v 0, U w = 0 in a left) neighborhood of T, w < 3 ) w w λ max U w < χt) 3 ϱ 0ϑ 0 χτ) 3 ϱ 0ϑ 0 Kt τ), t τ, T ]. Denoting w 0 = wτ, ) and shifting everything to the origin t = 0, we infer that there is a function w X 0,e 0, T ), with the following properties: w0, ) = w 0, et, x) = w 0 = χτ) 3 ϱ 0ϑ 0, wt, ) = v 0, 4.4) χτ) 3ϱ 0ϑ 0 Kt, t [ ] 0, χτ) χ 0 K χ 0 3 ϱ 0ϑ 0 for t [ χτ) χ 0 K, T ]. Similarly to 3.6), we introduce the set X 0, together with the function e[v] = χτ) 3 ϱ 0ϑ[v]. 4.5) Our ultimate goal is to show that the function w, introduced in 4.4), belongs to X 0 as long as we conveniently fix the parameters χ, K. To this end, it is enough to show that e, defined through 4.5), satisfies et, x) < e[w] = χτ) 3 ϱ 0ϑ[w] for all t 0, T ]. 4.6) For t ] 0, χτ) χ 0 K, this amounts to showing 3 ϱ 0ϑ[w] ϱ 0 ϑ 0 ) < Kt, t 0, χτ) χ ] 0, K which follows from 4.) provided K = Kχ) is taken large enough. Next, for t [ χτ) χ 0, T ], we have to check that K 3 ϱ 0ϑ[w] ϱ 0 ϑ 0 ) < χτ) [ ] χτ), t χ0, T, K which follows from 4.0), 4.3) provided we fix χ = χϱ 0, ϑ 0 ) large enough. Having found a suitable subsolution we can finish the proof of Theorem 4. exactly as in Section

27 5 Concluding remarks In this paper, we focused exclusively on the physically relevant 3D setting. The reader will have noticed that exactly the same results may be obtained also in the D case. Note, however, that the method does not apply to the D system as the conclusion of Lemma 3. is no longer available. Theorem 4. obviously applies to the larger class of dissipative solutions for which the internal energy balance is replaced by the entropy inequality )) ) ) ) ϑ 3/ ϑ 3/ x ϑ t ϱ log + div x ϱ log u div x xϑ. 5.) ϱ ϱ ϑ ϑ Moreover, we could even construct dissipative solutions with an artificial entropy production satisfying 5.) with strict inequality and, at the same time, conserving the total energy. On the other hand, a criterion based on maximality of the entropy production could be possibly used to identify a class of physically relevant solutions. The conclusion of Theorem 4. can be extended to the time interval [0, ) by means of continuation. Indeed we can take the function h in 3.5) such that ht ) = 0; whence Moreover, as pointed out in Remark 3. T, ) = ϱ 0. ϑt, ) [L p ), W,p )] α for all t [0, T ], therefore we can apply Theorem 4. recursively on the time intervals [nt, n + )T ], n=,... A similar extension of Theorem 4. seems possible but technically more complicated. Acknowledgment The authors would like to thank Camillo De Lellis for fruitful discussions about the problem. References [] T. Alazard. Low Mach number flows and combustion. SIAM J. Math. Anal., 384):86 3 electronic), 006. [] T. Alazard. Low Mach number limit of the full Navier-Stokes equations. Arch. Rational Mech. Anal., 80: 73,

28 [3] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis Friedrichroda, 99), volume 33 of Teubner-Texte Math., pages 9 6. Teubner, Stuttgart, 993. [4] S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math. ), 6):3 34, 005. [5] A. Bressan. Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. Oxford University Press, Oxford, 000. [6] E. Chiodaroli. A counterexample to well-posedness of entropy solutions to the compressible Euler system. 0. Preprint. [7] C. M. Dafermos. Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin, 000. [8] C. De Lellis and L. Székelyhidi, Jr. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal., 95):5 60, 00. [9] C. De Lellis and L. Székelyhidi, Jr. The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. N.S.), 493): , 0. [0] E. Feireisl. Relative entropies in thermodynamics of complete fluid systems. Discr. and Cont. Dyn. Syst. Ser. A, 3: , 0. [] E. Feireisl and A. Novotný. Singular limits in thermodynamics of viscous fluids. Birkhäuser- Verlag, Basel, 009. [] E. Feireisl and A. Novotný. Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Rational Mech. Anal., 04: , 0. [3] N. V. Krylov. Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal., 50):5 558, 007. [4] T. P. Liu. Admissible solutions of hyperbolic conservation laws. Mem. Amer. Math. Soc., 3040):iv+78, 98. [5] S. Müller and V. Šverák. Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. ), 573):75 74, 003. [6] D. Serre. Local existence for viscous system of conservation laws: H s -data with s > + d/. In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 56 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, 00. 6

29 Powered by TCPDF [7] D. Serre. The structure of dissipative viscous system of conservation laws. Phys. D, 395):38 386, 00. [8] A. Shnirelman. Weak solutions of incompressible Euler equations. In Handbook of mathematical fluid dynamics, Vol. II, pages North-Holland, Amsterdam, 003. [9] C. H. Wilcox. Sound propagation in stratified fluids. Appl. Math. Ser. 50, Springer-Verlag, Berlin,