TOPICAL PROBLEMS OF FLUID MECHANICS DOI: https://doi.org/.43/tpfm.27. NOTE ON THE PROBLEM OF DISSIPATIVE MEASURE-VALUED SOLUTIONS TO THE COMPRESSIBLE NON-NEWTONIAN SYSTEM H. Al Baba, 2, M. Caggio, B. Ducomet 3,. Ne asová Institute of Mathematics, itná 25, 5 67 Praha, Czech Republic 2 Laboratoire de Mathématiques et de leurs Applications, CNRS UMR 542, Université de Pau et des Pays de l'adour 3 CEA, DAM, DIF, F-9297 Arpajon, France Abstract We introduce a dissipative measure-valued solution to the compressible non-newtonian system. We generalized a result given by Novotný, Ne asová [4]. We derive a relative entropy inequality for measure-valued solution as an extension of the "classical" entropy inequality introduced by Dafermos [2], Mellet-Vasseur [], Feireisl-Jin-Novotný [5]. Formulation of the problem We consider measure-valued solutions of the compressible non-newtonian system. The advantage of measure-valued solutions is the property that in many cases, the solutions can be obtained from weakly convergent sequences of approximate solutions. Measure-valued solutions for systems of hyperbolic conservations laws were initially introduced by DiPerna [3]. He used Young measures to pass to limit in the arti cial viscosity term. In the case of the incompressible Euler equations, DiPerna and Majda [4] also proved global existence of measure-valued solutions for any initial data with nite energy. They introduced generalized Young measures to take into account oscillation and concentration phenomena. Thereafter the existence of measure-valued solutions was nally shown for further models of uids, e.g. compressible Euler and Navier-Stokes equations [3]. The measure-valued solution to the non-newtonian case was proved by Novotný, Ne asová [4]. Recently, weak-strong uniqueness for measure-valued solutions of isentropic Euler equations were proved in [8]. Inspired by previous results, the concept of dissipative measure-valued solution was nally applied to the barotropic compressible Navier-Stokes system [9]. It is a generalization of classical measure-valued solution. The motion of the uid is governed by the following system of equations t % + divx %u) = in, T ) Ω,.) t %u) + divx %u u) + x p = divx S in, T ) Ω,.2) where % is the mass density and u is the velocity eld, functions of the spatial position x R3 and the time t R. The scalar function p is termed pressure, given function of the density. In particular, we consider the isothermal case, namely p = λ%, with λ > a constant. The stress tensor is given by Sij = βul,l δij + 2ωei,j u),.3) where, ω=ω u b, divx u, det, β=β u b, divx u, det xj xj q uj u b = ei,j u)ei,j u), ei,j u) = +, 2 xj xi 2 β ω, 3 ω..4)
2 Prague, February 5-7, 27 We consider the Dirichlet boundary conditions and initial data u = in, T ).5) We consider the following hypothesis: 2ω û, div x u, det +β û, div x u, det u) = u, ϱ) = ϱ..6) 2ω û, div x u, det +β û, div x u, det û 2 div x u div x u k 2 û γ,.7) e ij u) div x u δ ij k û γ,.8) for i, j, 2, 3 with k, k 2 >, γ γ < γ +, γ 2. Further, we assume the existence of a positive function ϑe ij ) such that ϑ = 2ω û, div x u, det e ij u) + δ ij β û, div x u, det div x u..9) e ij Remark. We consider power-law type of uids. For more details see []). 2 Mathematical preliminaries We dene φt) = e t t and φ 2 t) = e t2 the Young functions and by ψt) = +t) ln + t) t, and ψ /2 t) the complementary Young functions to them. The corresponding Orlicz spaces are L φ ), L φ2 ), L ψ ), L ψ/2 ). These are Banach spaces equipped with a Luxembourg norm u Lf ) = inf h { h > ; ) } ux) f dx h < +, 2.) where f stands for φ, φ 2, ψ, ψ /2. Let C ) be the set of bounded continuous functions which are dened in. We denote C ψ, C φ, C ψ/2 and C φ2 the closure of C ) in L ψ ), L φ ), L ψ/2 ), and L φ2 ), respectively. We have C φ = L ψ ), C ψ = L φ ), C φ2 = L ψ/2 ), C ψ/2 ) ) = Lφ2 ), where C ψ, C φ, C ψ/2 and C φ2 are separable Banach spaces. Further, ψ, ψ /2 satisfy the 2 -condition and we have C ψ ) = L ψ ), C ψ/2 ) = L φ2 ). L w Q T, M R N 2 denotes the spaces of all weakly measurable mappings from Q T into M R N 2) R N 2 norm; ν L Q T, M R N 2 is a weakly measurable map i with nite L Q T, M x, t) ν x,t, gx, t is Lebesgue measurable in Q T for every g L Q T, C R N 2, N is dimension. We dene by L p ), W l,p ) resp. W l,p ), l, p < +, the usual Lebesque space, Sobolev spaces. We denote V k = W k,2 W,2, Q T =, T ). Remark 2. For more details about Orlicz spaces see [2]. Denition 3. measure-valued solution) Let ϱ, v, ν) be such that ϱ L I, L ψ ), 2.2) v L 2 I, V k ) L γ I, W,γ ), 2.3)
TOPICAL PROBLEMS OF FLUID MECHANICS 3 the functions are ν-integrable in R N 2, ν L w Q T, M R N 2, 2.4) σ ij, β σ, Trσ, det σ Trσ, ω σ, Trσ, det σ σ ij 2.5) R N2 σ ij dν t,x σ) = u i, a.e. in Q T. 2.6) Then, we dene a measured-valued solution for the system.)-.9) in the sense of DiPerna [3], in the following way ϕ i ϱu i Q T t dxdt ϱu i u j ϕ i,,j dxdt ϱ u ϕ i )dx k ϱϕ i,i dxdt Q T Q T ) + dxdt Q T β σ, Trσ, det σ Trσδ ij + 2ω σ, Trσ, det σ σ ij dν t,x σ) R N2 ϕ i,j = ϱb i ϕ i dxdt Q T 2.7) for all ϕ C Q t ), ϕt) W,γ ) for any t I, ϕt) =. Remark 4. In the Denition 3 the Young measures are dened for the gradient of the velocity eld. In the next Section the measures are considered for the density and the velocity eld. Theorem 5. Let u V k, ϱ C d ), ϱ > ε >, d =, 2,... Let assumptions.7) -.9) be satised, k > N. Then, there exists ϱ, u) and a family of a probability measure ν x,t on R N 2 with properties such that i) ν L w Q T, R N 2), ν x,t =, for a.e. x, t) Q T ; ii) supp ν x,t R N 2, for a.e. x, t) Q T ; ) iii) u L γ I, W,γ ) L γ I, W,α, αγ > N, α < ; iv) ϱ L I, L ψ L 2 I, W,2 ); v) ϱu L γ I, W α,γ ), αγ > N, α <, γ + γ = ; and such that ϱ, u, ν) satises 2.7). Proof. See [4]. vi) ϱu i u j L γ I, W α,γ )
4 Prague, February 5-7, 27 3 Dissipative measure-valued solutions to the compressible isothermal system We introduce the concept of dissipative measure-valued solution to the system.) -.2) in the spirit of [8] and [9]. Denition 6. We say that a parameterized measure {ν t,x } t,x),t ), ν L w, T ) ; P [, ) R N, ν t,x ; s ϱ, ν t,x ; v u is a dissipative measure-valued solution of the compressible Navier-Stokes system.) -.2) in, T ), with the initial conditions ν and dissipation defect D, D L, T ), D, if the following holds. i) Continuity equation. There exist a measure r C L [, T ], M and χ L, T ) such that for a.a. τ, T ) and every ψ C [, T ] ), and = τ ii) Momentum equation. r C τ) ; x ψ χ τ) D τ) ψ C ) 3.) ν t,x ; s ψ τ, ) dx ν ; s ψ, ) dx [ ν t,x ; s t ψ + ν t,x ; sv x ψ] dxdt + τ u = ν t,x ; v L 2, T ; W,2 ; R N, r C ; x ψ dt. 3.2) and there exists a measure r M L [, T ], M and ξ L, T ) such that for a.a. τ, T ) and every ϕ C [, T ] ; R N ), ϕ =, and = τ r M τ) ; x ϕ ξ τ) D τ) ϕ C ) 3.3) ν t,x ; sv ϕ τ, ) dx ν ; sv ϕ, ) dx [ ν t,x ; sv t ϕ + ν t,x ; s v v) : x ϕ + ν t,x ; ps) div x ϕ] dxdt τ τ S x v) : x ϕdxdt + r M ; x ϕ dt. 3.4) iii) Energy inequality. ν t,x ; 2 s u 2 + P s) +D τ) ) τ dx + S û, div x u, det ) ν ; 2 s u 2 + P s) : x udxdt dx, for a.e. τ, T ), where P s) = + s) ln + s) s. Moreover, the following version of Poincare's inequality holds τ ν t,x ; v u 2 dxdt cd τ).
TOPICAL PROBLEMS OF FLUID MECHANICS 5 Remark 7. Suppose that u ε u weakly in L 2, T ; W,p R N when ε, then τ ν t,x ; v u 2 τ dxdt = lim u ε u p dxdt ε τ τ c lim x u ε x u p dxdt = c lim x u ε p x u p dxdt ε ε cd τ) provided the dissipation defect D "contains" the oscillations and concentrations in the velocity gradient. We introduce the relative energy functional [ ] E ϱ, u r, U) = 2 ϱ u U 2 + P ϱ) P r)ϱ r) P r) dx, 3.5) P ϱ) = + ϱ) ln + ϱ) ϱ. In fact it is shown in [5] that any nite energy weak solution ϱ, u) to the compressible barotropic Navier-Stokes system satises the relative entropy inequality for any pair r, U) of suciently smooth test functions such that r > and U = and this inequality is an essential tool in order to prove the convergence to a target system. For other details see [6]. In the framework of dissipative measure-valued solution in the spirit of [9]-[8]) we dene the functional E mv ϱ, u, r, U) ν t,x ; 2 s v U 2 + P s) P r)ϱ r) P r) dx. Theorem 8. Let the parameterized measure {ν t,x } t,x),t ) with ν L w, T ) ; P [, ) R N, ν t,x ; s ϱ, ν t,x ; v u be a dissipative measure-valued solution to the compressible non-newtonian system.) -.2) with the initial condition ν and dissipation defect D. Then, s, ν) satises the following relative energy inequality τ E mv + S x u)eu) eu + Dτ) ν,x ; s v U, ) 2 2 ) + P s) P r )s r ) P r ) dx τ + Rs, v, r, U)t)dt 3.6) for a.a. τ, T ) and any pair of test functions r, U) such that U C [, T ], R n ),U =, r Cc Q T ), r >, where Rs, v, r, U)t) = ν t,x ; sv t U + ν t,x ; sv v x U dx + τ τ τ ν t,x ; ps) div x U) dx ν t,x ; s U t U + ν t,x ; sv U x U) dx ν t,x s r ) p r) t r ν t,x : sv p r) x rdx r r M ; x U τ dt + r C ; 2 x U 2 x P r) dx. 3.7)
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