Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system

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1 Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the orteweg system Frédéric Charve, Boris Haspot To cite this version: Frédéric Charve, Boris Haspot. Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the orteweg system. SIAM J. Math. Anal., 213, 45 2), pp <hal > HAL Id: hal Submitted on 26 Oct 211 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the orteweg system Frédéric Charve, Boris Haspot Abstract In the first part of this paper, we prove the existence of global strong solution for orteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the orteweg system converges in the case of a γ law for the pressure P ρ) = aρ γ, γ > 1) to entropic solution of the compressible Euler equations. In particular it justifies that the orteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in particular the classical theory of Lax and Glimm see [21, Lax,G 11]) can not be used. 1 Introduction We are concerned with compressible fluids endowed with internal capillarity. The model we consider originates from the XIXth century work by Van der Waals and orteweg VW,f [38, 22] and was actually derived in its modern form in the 198s using the second gradient theory, see for instance [9, fds,fjl,ftn 2, 37]. The first investigations begin with the Young-Laplace theory which claims that the phases are separated by a hypersurface and that the jump in the pressure across the hypersurface is proportional to the curvature of the hypersurface. The main difficulty consists in describing the location and the movement of the interfaces. Another major problem is to understand whether the interface behaves as a discontinuity in the state space sharp interface) or whether the phase boundary corresponds to a more regular transition diffuse interface, DI). The diffuse interface models have the advantage to consider only one set of equations in a single spatial domain the density takes into account the different phases) which considerably simplifies the mathematical and numerical study indeed in the case of sharp interfaces, we have to treat a problem with free boundary). Université Paris-Est Créteil, Laboratoire d Analyse et de Mathématiques Appliquées UM 85), 61 Avenue du Général de Gaulle, 94 1 Créteil Cedex France). frederic.charve@univ-paris12.fr Basque Center of Applied Mathematics, Bizkaia Technology Park, Building 5, E-4816, Derio Spain) Ceremade UM CNS 7534 Université de Paris IX- Dauphine, Place du Marchal DeLattre De Tassigny PAIS CEDEX 16, haspot@ceremade.dauphine.fr 1

3 Let us consider a fluid of density ρ, velocity field u, we are now interested in the following compressible capillary fluid model, which can be derived from a Cahn-Hilliard like free energy see the pioneering work by J.- E. Dunn and J. Serrin in [9] fds and also in fa,fc,fgp,hm [1, 3, 12, 17]). The conservation of mass and of momentum write: t ρɛ + x ρ ɛ u ɛ ) =, 1.1) 3systeme t ρɛ u ɛ ) + x ρ ɛ u ɛ ) 2 ) ɛ x ρ ɛ x u ɛ ) + x aρ ɛ ) γ ) = ɛ 2 x, where the orteweg tensor reads as following: div = x ρ ɛ κρ ɛ ) xx ρ ɛ κρɛ ) + ρ ɛ κ ρ ɛ )) x ρ ɛ 2) x κρ ɛ ) x ρ ɛ ) 2). 1.2) div κ is the coefficient of capillarity and is a regular function of the form κρ) = ɛ 2 ρ α with α. In the sequel we shall assume that κρ) = ɛ2 ρ. The term x allows to describe the variation of density at the interfaces between two phases, generally a mixture liquidvapor. P = aρ γ with γ 1 is a general γ law pressure term. ɛ corresponds to the controlling parameter on the amplitude of the viscosity and of the capillarity. When we set v ɛ = u ɛ + ɛ x ln ρ ɛ ), we can write 1.1) 3systeme on the following form we refer to [13] Hprepa for the computations): t ρɛ + x ρ ɛ v ɛ ) ɛ xx ρ ɛ =, t ρɛ v ɛ ) + x ρ ɛ u ɛ )v ɛ )) ɛ x ρ ɛ x v ɛ ) + x aρ ɛ ) γ ) =, We now consider the Cauchy problem of 1.3) 1.1 when the fluid is away from vacuum. Namely, we shall study 1.3) 1.1 with the following initial data: such that: 1.3) 1.1 ρ ɛ, x) = ρ ɛ x) >, u ɛ, x) = u ɛ x), 1.4) 1.2 lim x +, ρɛ x), u ɛ x)) = ρ +,, u +, ), with ρ +, >. We would like to study in the sequel the limit process of system 1.3) 1.1 when ɛ goes to and to prove in particular that we obtain entropic solutions of the Euler system: t ρ + xρv) =, 1.5) 1.3 t ρv) + xρv 2 ) + x aρ γ ) =, Let us now explain the interest of the capillary solutions for the hyperbolic systems of conservation laws. 1.1 Viscosity capillarity processes of selection for the Euler system In addition of modeling a liquid-vapour mixture, the orteweg also shows purely theoretical interests consisting in the selection of the physically relevant solutions of the 2

4 Euler model in particular when the system is not strictly hyperbolic). The typical case corresponds to a Van der Waals pressure: indeed in this case the system is not strictly hyperbolic in the elliptic region which corresponds to the region where the phase change occurs). In the adiabatic pressure framework P ρ) = ρ γ with γ > 1), the system is strictly hyperbolic and the theory is classical. More precisely we are able to solve the iemann problem when the initial Heaviside data is small in the BV space. Indeed we are in the context of the well known Lax result as the system is also genuinely nonlinear we refer to [21]). Lax It means we have existence of global C 1 -piecewise solutions which are unique in the class of the entropic solutions. This result as been extent by Glimm in the context of small initial data in the BV-space by using a numerical scheme and approximating the initial BV data by a C 1 -piecewise function which implies to locally solve the iemann problem via the Lax result). For the uniqueness of the solution we refer to the work of Bianchini and Bressan [2]) BB1 who use a viscosity method. In the setting of the Van der Waals pressure, the existence of global solutions and the nature of physical relevant solutions remain completely open. Indeed the system is not strictly hyperbolic anymore. If we rewrite the compressible Euler system in Lagrangian coordinates by using the specific volume τ = 1/ρ in 1 b, ) and the velocity u, the system satisfies in, + ) the equations: t τ x u =, t u x P 1.6) euler τ)) =, with the function P : 1 b, ), ) given by: P τ) = P 1 τ ), The two eigenvalues of the system are: λ 1 τ, v) = P τ), τ 1 b, ). λ 2 τ, v) = P τ). 1.7) vp The corresponding eigenvectors r 1, r 2 are: ) 1 w 1 τ, v) = P, w 2 τ, v) = τ) Furthermore by calculus we obtain: ) 1 P τ) 1.8) λ 1 τ, v) w 1 τ, v) = P τ) 2 P, λ 2 τ, v) w 2 τ, v) = τ) P τ) 2 P τ) 1.9) We now recall the definition of a standard conservation law in the sense of Lax it means entropy solutions): The system is strictly entropic if the eigenvalues are distinct and real. 3

5 The characteristics fields are genuinely nonlinear if we have for all τ, v), λ 1 τ, v) w 1 τ, v) and λ 2 τ, v) w 2 τ, v), for more details we refer to [33]. Serre The definition of genuine nonlinearity is some kind of extension of the notion of convexity to vector-valued functions in particular when we consider the specific case of the traveling waves). The previous assumptions aim at ensuring the existence and the uniqueness of the iemann problem see [1] Evans and [33]). Serre When P is a Van der Waals pressure, we observe that the first conservation law [33, Serre,Evans 1]) is far from being a standard hyperbolic system, indeed: It is not hyperbolic but elliptic) in 1 α 1, 1 α 2 ), the characteristic fields are not genuinely nonlinear in the hyperbolic part of the state space. Here the classical Lax-Glimm theory cannot be applied. In particular there doesn t exist any entropy-flux pair, which suggests that the entropy framework is not adapted for selecting the physically relevant solutions. In order to deal with this problem, Van der Waals and orteweg began by considering the stationary problem with null velocity, and solving P ρ) =. For more details we refer to [31]. ohdehdr It consists in minimizing in the following admissible set A = {ρ L 1 Ω)/W ρ) L 1 Ω), ρx)dx = m}, the following functionnal F [ρ] = Ω Ω W ρx))dx. Unfortunately this minimization problem has an infinity of solutions, and many of them are physically irrelevant. In order to overcome this difficulty, Van der Waals in the XIXth century was the first to regularize the previous functional by adding a quadratic term in the density gradient. More precisely he considered the following functional: Flocal ɛ = W ρ ɛ x)) + γ ɛ2 2 ρɛ 2) dx, with: Ω A local = H 1 Ω) A. This variational problem has a unique solution and its limit as ɛ goes to zero) converge to a physical solution of the equilibrium problem for the Euler system with Van der Waals pressure, that was proved by Modica in [28] EF with the use of gamma-convergence. By the Euler-Lagrange principle, the minimization of the Van der Waals functional consists in solving the following stationary problem: P ρ ɛ ) = γɛ 2 ρ ɛ ρ ɛ, where the right-hand side can be expressed as the divergence of the capillarity tensor. Heuristically, we also hope that the process of vanishing capillarity-viscosity limit selects the physical relevant solutions as it does for the stationary system. This problem actually remains open. 4

6 1.2 Existence of global entropic solutions for Euler system Before presenting the results of this paper let us recall the results on this topic in these last decades. We shall focus on the case of a γ pressure law P ρ) = aρ γ with γ > 1 and a positive. Let us mention that these cases are the only ones well-known essentially because the system is strictly hyperbolic in this case and that we can exhibit many entropy-flux pairs). Here the Lax-Glimm theory can be applied, however at the end of the 7 s, one was interested in relaxing the conditions on the initial data by only assuming ρ and u in L. In the beginning of the 8 s Di Perna initiated this program, consisting in obtaining global entropic solutions for L initial data. Di1, Di2 Indeed in [7, 8], Di Perna prove the existence of global weak entropy solution of 1.5) 1.3 for γ = d+1 and γ = 2k + 2k + 1 with k 1), d 2 by using the so-called compensated compactness introduced by Tartar in [35]. Ta This result was extended by Chen in [4] Chen in the case γ 1, ] and by Lions et al in [26] in the case γ [3, ). In [25], 36 Lions et al generalize this result to the general case γ 1, 3), and finally the case γ = 1 is treated by [18]. Hu1 We would like to mention that these results are obtained through a vanishing artificial viscosity on both density and velocity. The problem of vanishing physical viscosity limit of compressible Navier-Stokes equations to compressible Euler equations was until recently an open problem. However Chen and Perepelista in [5] 1 proved that the solutions of the compressible Navier-Stokes system with constant viscosity coefficients converge to a entropic solution of the Euler system with finite energy. This result was extended in [19] Hu2 to the case of viscosity coefficients depending on the density. Inspired by [5] 1 and [19], Hu2 we would like to show that the solution of the orteweg system 1.3) 1.1 converges to a entropic solution of the Euler system with finite energy when the pressure is a γ law. To do this, we will prove for the first time up our knowledge the existence of global strong solution for the orteweg system in one dimension in the case of Saint-Venant viscosity coefficients. By contrast, the problem of global strong solutions for compressible Navier-Stokes equations remains open indeed one of the main difficulties consists in controlling the vacuum). This result justifies that the orteweg system allows us to select the relevant physical solutions of the compressible Euler system at least when the pressure is adiabatic P ρ) = aρ γ with γ > 1). The problem remains open in the case of a Van der Waals pressure. 1.3 esults Let us now describe our main result. In the first theorem we prove the existence of global strong solution for the orteweg system ). Theorem 1.1 Let ρ >. Assume that the initial data ρ and u satisfy: < m ρ M < +, ρ ρ H 1 ), v H 1 ) L ). 1.1) 2.5 Then there exists a global strong solution ρ, v) of 1.3) 1.1 on + such that for every T > : ρ ρ L, T, H 1 )), ρ L, T, L )), v L, T, H 1 )) L 2, T, H 2 )) and v L, T, L )). 5

7 theo theo1 theo2 Finally this solution is unique in the class of weak solutions satisfying the usual energy inequality. emark 1 We would like to point out that the problem remains open in the case of the Saint-Venant system, which corresponds to system ) without capillarity. In the following theorem, we are interested in proving the convergence of the global solutions of system 1.3) 1.1 to entropic solutions of the Euler system 1.5). 1.3 Theorem 1.2 Let γ > 5 3 and ρɛ, v ɛ ) with m ɛ = ρ ɛ v ɛ be the global solution of the Cauchy problem 1.3) 1.1 with initial data ρ ɛ, vɛ ) as in theorem theo 1.1).Then, when ɛ, there exists a subsequence of ρ ɛ, m ɛ ) that converge almost everywhere to a finite entropy solution ρ, ρv) to the Cauchy problem 1.5) 1.3 with initial data ρ, ρ v ). emark 1 We would like to point out that Lions et al in [25] 36 had obtained the existence of global entropic solution for γ > 1 by a viscosity vanishing process, and the considered regularizing system was exactly the orteweg system modulo the introduction of the effective velocity. One important basis of our problem for theorem 1.2 theo1 is the following compactness theorem established in [5]. 1 Theorem 1.3 Chen-Perepelitsa [5]) 1 Let ψ C 2), ηψ, q ψ ) be a weak entropy pair generated by ψ. Assume that the sequences ρ ɛ x, t), v ɛ x, t)) defined on + with m ɛ = ρ ɛ v ɛ, satisfies the following conditions: 1. For any < a < b < + and all t >, it holds that: t b where Ct) > is independent of ɛ. a ρ ɛ ) γ+1 dxdτ Ct, a, b), 1.11) For any compact set, it holds that t ρ ɛ ) γ+θ + ρ ɛ v ɛ 3) dxdτ Ct, ), 1.12) 1.9 where Ct, ) > is independent of ɛ. 3. The sequence of entropy dissipation measures η ψ ρ ɛ, m ɛ ) t + q ψ ρ ɛ, m ɛ ) x are compact in H 1 loc 2 +). 1.13) 1.1 Then there is a subsequence of ρ ɛ, m ɛ ) still denoted ρ ɛ, m ɛ )) and a pair of measurable functions ρ, m) such that: ρ ɛ, m ɛ ) ρ, m), a.e as ɛ. 1.14) 1.11 emark 2 We would like to recall that the estimate 1.12) 1.9 was first derived by Lions et al in [26] 35 by relying the moment lemma introduced by Perthame in [3]. Per The paper is arranged as follows. In section 2 section2 we recall some important results on the notion of entropy enrtopy-flux pair for Euler system and on the kinetic formulation of Lions et al in [26]. 35 In section 3, section3 we show theorem 1.1 theo and in the last section 4.1 section4 we prove theorem 1.2. theo1 6

8 section2 2 Mathematical tools Definition 2.1 A pair of functions ηρ, v), Hρ, v)) or ηρ, m), qρ, m)) for m = ρv, is called an entropy-entropy flux pair of system ), if the following holds: [ηρ, v)] t + [Hρ, v)] x =, for any smooth solution of ). Furthermore ηρ, v) is called a weak entropy if: η, u) =, for any fixed v. Definition 2.2 An entropy ηρ, m) is convex if the Hessian 2 ηρ, m) is nonnegative definite in the region under consideration. Such η satisfy the wave equation: tt η = θ 2 ρ γ 3 xx η. From [26], 35 we obtain an explicit representation of any weak entropy η, q) under the following form: η ψ ρ, m) = χρ, s v)ψs)ds, 2.15) 1.4 H ψ ρ, m) = θs + 1 θ)u)χρ, s v)ψs)ds, where the kernel χ is defined as follows: χρ, v) = [ρ 2θ v 2 ] λ +, λ = 3 γ 2γ 1) > 1 2, and θ = γ 1, 2 and here: Proposition 2.1 see 35 [26]) t λ + =t λ for t >, = for t, For instance, when ψs) = 1 2 s2, the entropy pair is the mechanical energy and the associated flux: η ρ, m) = m2 2ρ + eρ), q ρ, m) = m3 2ρ 2 + e ρ), 2.16) 1.5 where eρ) = κ γ 1 ργ represents the gas internal energy in physics. In the sequel we will work far away of the vacuum that it why we shall introduce equilibrium states such that we avoid the vacuum. Let ρx), vx)) be a pair of smooth monotone functions satisfying ρx), vx)) = ρ,+, v,+ ) when + x L for some large L >. The total mechanical energy for 1.3) 1.1 in with respect to the pair of reference function ρx), vx)) is: 1 E[ρ, v]t) = 2 ρt, x) vt, x) vx) 2 + e ρt, x), ρx)) ) dx 2.17) 1.7 7

9 where e ρ, ρ) = eρ) e ρ) e ρρ ρ). The total mechanical energy for system??) 3system with κρ) = κ ρ is: E 1 [ρ, u]t) = 1 2 ρt, x) ut, x) ūx) 2 + e ρt, x), ρx)) + ɛ 2 x ρ 1 2 ) 2 ) dx 2.18) 1.7 and the total mechanical energy for system 1.3) 1.1 is: 1 E 2 [ρ, v]t) = 2 ρt, x) vt, x) vx) 2 + e ρt, x), ρx)) ) dx 2.19) 1.7 Definition 2.3 Let ρ, v ) be given initial data with finite-energy with respect to the end states: ρ ±, v ± ) at infinity, and E[ρ, v ] E < +. A pair of measurable functions ρ, u) : is called a finite-energy entropy solution of the Cauchy problem 1.5) 1.3 if the following properties hold: 1. The total energy is bounded in time such that there exists a bounded function CE, t), defined on + + and continuous in t for each E + with for a.e t > : E[ρ, v]t) CE, t). 2. The entropy inequality: η ψ ρ, v) t + q ψ ρ, v) x, is satisfied in the sense of distributions for all test functions ψs) {±1, ±s, s 2 }. 3. The initial data ρ, v ) are obtained in the sense of distributions. We now give our main conditions on the initial data ), which is inspired from 1 [5]. Definition 2.4 Let ρx), vx)) be some pair of smooth monotone functions satisfying ρx), vx)) = ρ,+, v,+ ) when +x L for some large L >. For positive constant C, C 1 and C 2 independent of ɛ, we say that the initial data ρ ɛ, vɛ ) satisfy the condition H if they verify the following properties: ρ ɛ >, ρɛ x) uɛ x) ūx) C < +, The energy is finite: 1 2 ρɛ x) vx) ɛ vx) 2 + e ρ ɛ x), ρx)) ) dx C 1 < +, ɛ 2 x ρ ɛ x) 2 ρ ɛ x)3 2α dx C 2 < +. In this section, we would like to recall some properties on the pair of entropy for the system 1.5). 1.3 Smooth solutions of 1.5) 1.3 satisfy the conservation laws: t ηρ, u) + x Hρ, u) =, 8

10 if and only if: η ρρ = P ρ) ρ 2 η uu. 2.2) ondes We supplement the equation 2.2 ondes by giving initial conditions: η, u) =, η ρ, u) = ψu). 2.21) initial We are now going to give a sequel of proposition on the properties of η, we refer to 35 [26] for more details. proputile Proposition 2.2 For ρ, u, ω, The fundamental solution of 2.2)- ondes 2.21) initial is the solution corresponding to η ρ, u) = δu) is given by: χρ, ω) = ρ γ 1 ω 2 ) λ + with λ = 3 γ 2γ 1). 2.22) The solution of 2.2)- ondes 2.21) initial is given by: ηρ, u) = ψξ)χρ, ξ u)dξ, 2.23) entropie η is convex in ρ, ρu) for all ρ, u if and only if g is convex. The entropy flux H associated with η is given by: Hρ, u) = ψξ)[θξ + 1 θ)ξ]χρ, ξ u)dξ where θ = γ ) flux We now give a important result on the entropy pair see 35 [26], lemma 4). pair35 Proposition 2.3 Taking ψs) = 1 2s s, then there exists a positive constant C >, depending only on γ > 1, such that the entropy pair η ψ, H ψ ) satisfies: η ψ ρ, u) ρ u 2 + ρ γ ), H ψ ρ, u) C 1 ρ u 3 + ρ γ+θ ), for all ρ and u, ηmρ, ψ u) ρ u + ρ θ ), ηmmρ, ψ u) Cρ ) 2.37 We are now going to give recent results on the entropy pair η ψ, q ψ ) generated by ψ 1 ) we refer to [5] for more details). C 2 propchen Proposition 2.4 For a C 2 function ψ :, compactly supported on the interval [a, b], we have: suppη ψ ), suppq ψ ) {ρ, m) = ρ, ρu) : u + ρ θ a, u ρ θ b} : 2.26) 3.2 Furthermore, there exists a constant C ψ such that, for any ρ and u, we have: 9

11 For γ 1, 3], η ψ ρ, m) + q ψ ρ, m) C ψ ρ. 2.27) 3.3 For γ 3, + ), η ψ ρ, m) C ψ ρ, q ψ ρ, m) C ψ ρ + ρ θ+1 ). 2.28) 3.4 If η ψ is considered as a function of ρ, m), m = ρu then η ψ mρ, m) + ρη ψ mmρ, m) C ψ, 2.29) 3.5 and, if η ψ m is considered as a function of ρ, u), then η ψ mm, u) + ρ 1 θ η ψ mρρ, ρu) C ψ. 2.3) 3.6 We now would like to express the kinetic formulation of ) introduced in 35 [26]). Theorem 2.4 Let ρ, ρv) L +, L 1 )) have finite energy and ρ, then it is an entropy solution of 1.5) 1.3 if and only if there exists a non-positive bounded measure m on + 2 such that the function χρ, ξ u) satisfies: t χ + x [θξ + 1 θ)u)χ] = ξξ mt, x, ξ). 2.31) cinetique section3 3 Proof of theorem theo 1.1 We would like to start with recalling an important result due to Solonnikov see Sol [34]). Let ρ the initial density such that: When the viscosity coefficient µρ) satisfies: < m ρ M < ) initiald µρ) c > for allρ, 3.33) visco we have the existence of strong solution for small time. More exactly, we have: Proposition 3.5 Let ρ, v ) satisfy 3.32) initiald and assume that µ satisfies 3.33), visco then there exists T > depending on m, M, ρ ρ H 1 and v H 1 such that 1.3) 1.1 has a unique solution ρ, v) on, T ) satisfying: for all T 1 < T. ρ ρ L H 1 ), t ρ L 2, T 1 ) ), v L 2, T 1, H 2 )), t v L 2, T 1 ) ) emark 3 The main point in this theorem is that the time of existence T depends only of the norms of ρ which gives us a low bounds on T of the system 1.3)

12 In view of this proposition, we see that if we introduce a truncated viscosity coefficient µ n ρ): µ n ρ) = maxρ, 1 n ), then there exists approximated solutions ρ n, v n ) defined for small time, T ) of the system 1.3). 1.1 In order to prove theorem 1.1 theo, we only have to show that ρ n, v n ) satisfies the following bounds uniformly with respect to n and T large: < m ρ n M < +, t [, T ], ρ n ρ L T H 1 )), v n L T H 1 )). 3.34) We are going to follow the method of Lions et al in [25], 36 indeed the main point is to prove that we can extend the notion of iemann invariant or more precisely the kinetic formulation of proposition 2.4 cinetique to the system 1.3). 1.1 We recall that system 1.3) 1.1 has the following form: t ρ n + x ρ n v n ) ɛ xx ρ n =, 3.35) 1.1a t ρ nv n ) + x ρ n v n v n ) ɛ x x ρ n v n ) ɛ x ρ n x v n ) + x aρ n ) γ =, and we have finally: t ρ n + x ρ n v n ) ɛ xx ρ n =, t ρ nv n ) + x ρ n v n v n ) ɛ x x ρ n v n ) + x aρ n ) γ =, 3.36) 1.1b Following [25] 36 and setting m n = ρ n v n we have for any pair of entropy flux ηρ, u), Hρ, u)) defined by 2.23) entropie and 2.24) flux where η is a convex function of ρ n, m n ). We write η = ηρ n, m n ): t η + x H = ɛ η ρ xx ρ n + ɛ η m xx m n, Here we define µ n such that: = ɛ xx η ɛ η ρρ x ρ n ) η ρm x ρ n ) x m n ) + η mm x m n ) 2 ). µ n = η ρρ x ρ n ) η ρm x ρ n ) x m n ) + η mm x m n ) 2 By proposition proputile 2.2, we can check that µ n. We obtain then that: t ηρ n, v n ) + x Hρ n, v n ) ɛ η ρ xx ρ n in, + ). By applying the same method than for proving the theorem cinetique 2.4, we obtain the following kinetic formulation: t χ + x [θξ + 1 θ)v n ]χ) xx χ = ξξ m n on 2, + ), 3.37) riemann 11

13 where m n is a nonpositive bounded measure on 2, + ). Finally we recover the classical maximum principle by multiplying 3.37) riemann by the convex functions gξ) = ξ ξ ) + and gξ) = ξ ξ ) and integrating over 2, + ). Indeed as we have that: and that: For ξ large enough, we can show that: We have obtain then that: C minv ρ θ x ) max v + ρ θ x ) C, suppξ = [v ρ θ, v + ρ[θ]. suppξ suppχ =. C minv ρ θ x ) v n ρ θ n v n + ρ θ n max v + ρ θ x ) C. In particular we obtained that ρ n and v n are uniformly bounded in L, T n, L )) or: ρn t, x) + v n t, x) ) C, 3.38) imp2 sup x,t,t n) section4 4 Proof of theorem 1.2 theo1 4.1 Uniform estimates for the solutions of ) First we assume that ρ ɛ, v ɛ ) is the global solutions of orteweg s equations 1.3) 1.1 constructed in theorem 1.1 theo and satisfying: and ρ ɛ t, x) c ɛ t), for some c ɛ t) >, 4.39) 2.1 lim x ± ρɛ, v ɛ )x, t) = ρ ±, u ± ). 4.4) 2.2 Here we are working around a non constant state ρ, v) with: lim ρ, v)x, t) = x ± ρ±, u ± ). It is a simple extension of theorem 1.1. theo Our goal is now to check the properties 1.11), ) 1.9 and 1.13) 1.1 in order to use the theorem 1.3 theo2 of Chen and Perepelista see [5]) 1 in order to prove the theorem 1.2. theo1 For simplicity, throughout this section, we denote ρ, v) = ρ ɛ, v ɛ ) and C > denote the constant independent of ɛ. We start with recalling the inequality energy for system 1.3), 1.1 indeed by the introduction of the effective velocity we obtain new entropies see [13]). Hprepa Lemma 1 Suppose that E 1 [ρ, u ] E < + for some E > independent of ɛ. It holds that: t sup E 1 [ρ, u]τ) + ɛ ρu 2 xdxdτ Ct), 4.41) 2.3a τ t 12

14 and: sup E 2 [ρ, v]τ) + ɛ τ t t ρv 2 xdxdτ + ɛ t ρ γ 2 ρ 2 xdxdτ Ct), 4.42) 2.3 lemma1 where Ct) depends on E, t, ρ, and ū but not on ɛ. Proof: It suffices to writes the energy inequalities for system 1.3) 1.1 and from system 1.5). 1.3 More exactly we have: d ηρ, m) η ρ, ρū)dx + ɛ ρu 2 dt xdx = qρ, m ) qρ +, m + ), with the entropy pair: with eρ) = ηρ, m) = m2 2ρ a γ 1 ργ. Since we have: + eρ), qρ, m) = m3 2ρ 2 + me ρ), ɛρ, ρ) ρρ θ ρ θ ) 2, θ = γ 1, 2 we can classically bootstrap on the left hand-side the term qρ, m ) qρ +, m + ). emark 4 Since vacuum could occur in our solution, the inequality t ρu 2 xdxdτ Ct), in 4.42) 2.3 is much weaker than the corresponding one in [5]. 1 That is why lemma 2 lemme2 will be more tricky to obtain. lemme2 The following higher order integrability estimate is crucial in compactness argument. Lemma 2 If the conditions of lemma 1 lemma1 hold, then for any < a < b < + and all t >, it holds that: t b where Ct) > depends on E, a, b, γ, t, ρ, ū but not on ɛ. a ρ γ+1 dxdτ Ct, a, b), 4.43) 2.21 emark 5 The proof follows the same ideas than in the case of compressible Navier- Stokes equations when we wish to obtain a gain of integrability on the density. We refer to [24] fl2 for more details. The proof is also inspired from Huang et al in [19]. Hu2 Proof. Choose ω C ) such that: ωx) 1, ωx) = 1 for x [a, b], and suppω = a 1, b + 1). By the momentum equation of ) and by localizing, we have P ρ)ω) x = ρuvω) x + P ρ) + ρuv)ω x ρv) t ω + ɛρv x ω) x ɛρv x ω x. 4.44)

15 Integrating 4.44) 2.22 with respect to spatial variable over, x), we obtain: x x P ρ)ω = ρuvω + ɛρv x ω) x ρv ωdy) t + [ρuv + P ρ))ω x ɛρv x ω x. 4.45) 2.23 Multiplying 4.45) 2.23 by ρω, we have x ρp ρ)ω 2 = ρ 2 uvω 2 + ɛρ 2 v x ω 2 ρω ρu) x ω x =ɛρ 2 v x ω 2 ρω + ρuω x x x ρu ωdy) + ρω ρv ωdy + ρω x ρv ωdy) t ρuω x ρv ωdy) t We now integrate ) over, t) and we get: t + ρ ω aρ γ+1 ω 2 dxdτ = ɛ x t + t ρ v ωdy)dx + ρω x ρ 2 v x ω 2 t [ρuv + P ρ))ω x ɛρv x ω x ]dx, x ρv ωdy) x [ρuv + P ρ))ω x ɛρv x ω x ]dx, ρuωx x x ρω ρv ωdy)dx ρv ωdy ) dxdτ [ρuv + P ρ))ω x ɛρv x ω x ]dx ) dxdτ. 4.46) ) 2.25 Let A = {x : ρt, x) ρ}, where ρ = 2 maxρ+, ρ ), 4.48) 2.26 then we have the following estimates by 4.42): 2.3 A Ct) e = dt). 4.49) ρ, ρ) By ), for any t, x) there exists a point x = x t, x) such that x x dt) and ρt, x ) = ρ. Here we choose β = γ+1 2 >, supp x suppω) ɛρ β t, x) ɛρ β + supp x suppω) A ɛρ β t, x), 2ɛρ β + supp x suppω) A ɛρ β t, x) ɛρ β t, x ), x 2ɛρ β +dt) + supp x suppω) A β ɛρ β 1 t, x)ρ x dx, 2ɛρ β + Ct) + Ct). b+1+2dt) a 1 2dt) b+1+2dt) a 1 2dt) x dt) β ρ 2β 1 dx + ρ γ dx, ɛ 2 ρ 1 ρ 2 xdx, 4.5)

16 Using 4.5), 2.28 Young inequalities and Hölder s inequalities, the first term on the right hand side of 4.47) 2.25 is treated as follows: t ρ 2 v x ω 2 dxdτ 1 t 2 ɛ ρ 3 ω 4 dxdτ + 1 t 2 ɛ ρvxdxdτ, 2 t Ct) + ɛ ρ 3 ω 2 dxdτ, 4.51) 2.29 t Ct) + Ct) ρ 4 β ω 2 dxdτ, t Ct) + δ ρ γ+1 ω 2 dxdτ, Here we have used the fact that γ > 5 lemma1 3. By lemma 1 and the Hölder inequality, we obtain x ρvωdy ρv dy, Then: ρω x Similarly, we have: and suppω) ρvωdy ) dx + t ɛ t ɛ ɛ t + ρω x t ρω x suppω) t ρdy) 1 2 ρ ω x ρuωx x suppω) ρ v ωdy ) dx ρv 2 dy) 1 2 Ct). 4.52) 2.3 ρvωdy ) dxdτ Ct). 4.53) 2.32 ρuv + P ρ))ω x dy ) dxdτ Ct), 4.54) 2.33 ρω ρv x ω x dy ) dxdτ ρ v x ω x dy ) dxdτ, ρωxdy 2 ) dτ, ρωdx ) ρvxdy 2 + Ct). Substituting 4.51), ) ) 2.34 into 4.47) 2.25 and noticing the smallness of δ, we proved lemma 2. lemme2 Lemma 3 Suppose that ρ x), v x) satisfy the conditions in the lemmas 1. lemma1 Furthermore there exists M > independent of ɛ, such that ρ x) v x) vx) dx M < +, 4.56) ) 2.34

17 then for any compact set, it holds that: t ρ γ+θ + ρ v 3 )dxdτ Ct, ), 4.57) 2.36 where Ct, ) is independent of ɛ. emark 6 In order to prove the inequality 4.57), 2.36 we will use the same ingredients than in [26] 35 where this inequality was obtained for the first time. Proof. We are now working with the function ψ of proposition 2.3. pair35 If we consider ηm ψ as a function depending of ρ, v), we have for all ρ and v : { η ψ mv ρ, v) C, ηmρρ, ψ v) Cρ θ ) For this weak entropy pair η ψ, H ψ ), we observe that: η ψ ρ, ) = η ψ ρ ρ, ) =, H ψ ρ, ) = θ 2 ρ3θ+1 and: By Taylor formula, we have: with: ηmρ, ψ ) = βρ θ with β = s [1 s 2 ] λ +ds. s 3 [1 s 2 ] λ +, η ρ, m) = βρ θ m + rρ, m), 4.59) 2.39 rρ, m) Cρv 2, 4.6) 2.4 for some constant C >. Now we introduce a new entropy pair η, Ĥ) such that, ηρ, m) = η ψ ρ, m ρv ), Ĥρ, m) = H ψ ρ, m ρv ) + v η ψ ρ, m ρv ), with m = ρv which satisfies: { ηρ, m) = βρ θ+1 v v ) + rρ, ρv v )), rρ, ρv v )) Cρv v ) 2. Integrating ) 1 η ρ ) 2 η m over, t), x), we have: 4.61) 2.41 x ηρ, m) ηρ, m ) ) t dy + q ρ, ρv v )) + v η ρ, ρv v ))dτ = tq ρ, ) + ɛ t By using 4.58), 2.38 we obtain: ɛ t x η m ρv x dτ ɛ t x η mu ρv 2 xdydτ Cɛ 16 η mu ρv 2 x + η mρ ρρ x v x )dydτ. t 4.62) 2.42 ρv 2 xdy dτ Ct), 4.63) 2.43

18 ɛ t x η mρ ρρ x v x dydτ Cɛ Cɛ t t ρv 2 xdy dτ + Cɛ ρ θ 1 ρ ρ x v x dy dτ Ct), t ρ γ 2 ρ 2 xdy dτ Ct). 4.64) 2.44 Substituting 4.63) 2.43 and 4.64) 2.44 into 4.62), 2.42 then integrating over and using 2.25), 2.37 we obtain: t ρ θ+γ + ρ v v 3 dxdτ t t Ct) + C η ρ, ρv v ) dxdτ + Cɛ ρ v v x dxdτ 4.65) 2.45 t + Cɛ ρ 1+θ x v x dxdτ + 2 sup êtaρy, τ), ρv)y, τ))dy)dx. τ [,t] Applying lemma lemma1 1, we have: t By Hölder s inequality and ), we get: ɛ t We have now: ɛ t η ρ, ρv v ) dxdτ Ct). 4.66) 2.46 t ρ 1+θ v x dxdτ Cɛ ρvxdxdτ 2 + Cɛ t t Ct) + Ct) ρ θ dxdτ, Ct). ρ v v x dxdτ 1 t 2 ɛ Ct). ρvxdxdτ t 2 ɛ ρ 1+2θ dxdτ, ρv 2 dxdτ, 4.67) ) 2.48 Now we are going to deal with the last term on the right hand side of 4.65) ) 1.1 implies that: ρv ρv ) t + ρv 2 + P ρ) ρuu ) x = ɛρv x ) x. 4.69) 2.49 Integrating 4.69) 2.49 over [, t], x) for x, we get: x ρv ρv )dy = x ρ v ρv )dy t ρv 2 + P ρ) ρuu P ρ )) + ɛ t ρv x dτ. 4.7)

19 Furthermore: x x ηρy, τ), ρv)y, τ))dy x ηρρv) βρ θ+1 v v))dy + rρρv v))dy + Ct) + βρ ) θ x x ρv v))dy. x βρ θ+1 v v))dy βρ θ ρ ) θ )ρv v))dy + βρ ) θ x ρv v))dy, By using 4.56), 2.35 lemma 1 lemma1 and 2, lemme2 4.7) 2.5 and 4.71) 2.51 we conclude the proof of the lemma. 4.2 H 1 loc 2 +) Compactness 4.71) 2.51 In this section we are going to take profit of the uniform estimates obtained in the previous section in order to prove the following lemma, which gives the H 1 loc 2 +)-compactness of the orteweg solution sequence ρ ɛ, v ɛ ) on a entropy- entropy flux pair. Lemma 4 Let ψ C 2 ), ηψ, H ψ ) be a weak entropy pair generated by ψ. Then for the solutions ρ ɛ, v ɛ ) with m ɛ = ρ ɛ v ɛ of orteweg system , the following sequence: η ψ ρ ɛ, m ɛ ) t + q ψ ρ ɛ, m ɛ ) x are compact in H 1 loc 2 +) 4.72) 3.1 lemme4 Proof: Now we are going to prove the lemma. A direct computation on 1.3) ηρ ψ ρ ɛ, m ɛ ) + 1.3) ηmρ ψ ɛ, m ɛ ) gives: ηρ ψ ρ ɛ, m ɛ ) t + Hρ ψ ρ ɛ, m ɛ ) x = ɛ ηρ ψ ρ ɛ, m ɛ )ρ ɛ )vx ɛ ) ɛη ψ mu ρ ɛ, m ɛ )ρ ɛ )vx) ɛ 2 ɛη ψ muρ ɛ, m ɛ )ρ ɛ )v ɛ xρ ɛ x. 4.73) 3.7 Let be compact, using proposition 2.4 propchen 2.3) 3.6 and Hölder inequality, we get: t ɛ ηmuρ ψ ɛ, m ɛ )ρ ɛ ) vx) ɛ 2 + ηmuρ ψ ɛ, m ɛ )ρ ɛ )vxρ ɛ ɛ x dxdt t t Cɛ ρ ɛ ) vx) ɛ 2 dxdτ + Cɛ ρ ɛ ) γ 2 ρ ɛ x) ) 3.8 dxdτ This shows that: Ct). ɛη ψ muρ ɛ, m ɛ )ρ ɛ v ɛ x) 2 ɛη ψ muρ ɛ, m ɛ )ρ ɛ v ɛ xρ ɛ x are bounded in L 1 [, T ] ), 4.75) 3.9 and thus it is compact in W 1,p 1 loc 2 +), for 1 < p 1 < 2. Moreover we observe that η ψ muρ ɛ, ρ ɛ v ɛ ) C ψ, 18

20 , then we obtain: t ɛηmρ ψ ɛ, m ɛ )ρ ɛ vx) ɛ 4 3 dxdt t ɛ 4 3 ρ ɛ 4 3 v ɛ x 4 3 dxdt t Cɛ 4 3 ρ ɛ vx ɛ 2 dxdt + Cɛ 4 3 Ct, )ɛ Cɛ 4 3 Using ) and ), we obtain t t ρ ɛ ) 2 dxdt ρ ɛ ) γ+1 dxdt ɛ. 4.76) 3.1 ηρ ψ ρ ɛ, m ɛ ) t + Hρ ψ ρ ɛ, m ɛ ) x are compact in W 1,p 1 loc 2 +) for some 1 < p 1 < ) 3.11 Furthermore by 2.27) ), 3.4 lemma 1- lemma1 2 lemme2 and 4.57), 2.36 we have: η ψ ρ ρ ɛ, m ɛ ) t + H ψ ρ ρ ɛ, m ɛ ) x are uniformly bounded in L p 3 loc 2 +) for p 3 > ) 3.12 where p 3 = γ + 1 > 2 when γ 1, 3], and p 3 = γ+θ 1+θ we conclude the proof of the lemma 4. lemme4 > 2 when γ > 3. By interpolation 5 Proof of theorem 1.2 theo1 From lemmas 1, lemma1 we have verified the conditions i)-iii) of theorem 1.3 theo2 for the sequence of solutions ρ ɛ, m ɛ ). Using theorem 1.3, theo2 there exists a subsequence ρ ɛ, m ɛ ) and a pair of measurable functions ρ, m) such that ρ ɛ, m ɛ ) ρ, m), a.e ɛ. 5.79) 4.1 It is easy to check that ρ, m) is a finite-energy entropy solution ρ, m) to the Cauchy problem 1.5) 1.3 with initial data ρ, ρ u ) for the isentropic Euler equations with γ > 5 3. It achieves the proof of theorem 1.2. theo1 eferences fa BB1 fc Chen [1] D.M. Anderson, G.B McFadden and A.A. Wheller. Diffuse-interface methods in fluid mech. In Annal review of fluid mechanics, Vol. 3, pages Annual eviews, Palo Alto, CA, [2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math., ), pp [3] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys ) [4] G. Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, Lecture notes, Preprint MSI , Berkeley, October

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