Low Mach number limit of the full Navier-Stokes equations

Transcription

1 Low Mach number limit of the full NavierStokes equations Thomas Alazard Mathématiques Appliquées de Bordeaux Université Bordeaux cours de la Libération Talence FRANCE thomas.alazard@math.ubordeaux.fr / Porquerolles France p.1/14

2 / Porquerolles France p.2/14 Governing equations Full adimensioned NavierStokes equations:

3 / Porquerolles France p.2/14 Governing equations Full adimensioned NavierStokes equations: with parameters

4 / Porquerolles France p.2/14 Governing equations Full adimensioned NavierStokes equations: with parameters the variables are

5 / Porquerolles France p.3/14 Structural Assumptions

6 Structural Assumptions We choose to work with the pressure temperature. The system is supplemented with two state laws such that 1. existence of an entropy : ; 2. positivity of the coefficients of isothermal compressibility thermal expansion of the specific heats: / Porquerolles France p.3/14

7 Structural Assumptions We choose to work with the pressure temperature. The system is supplemented with two state laws such that 1. existence of an entropy : ; 2. positivity of the coefficients of isothermal compressibility thermal expansion of the specific heats: Example : ). / Porquerolles France p.3/14

8 Structural Assumptions We choose to work with the pressure temperature. The system is supplemented with two state laws such that 1. existence of an entropy : ; 2. positivity of the coefficients of isothermal compressibility thermal expansion of the specific heats: Example : ). Motivation : Low Mach number combustion. / Porquerolles France p.3/14

9 Setting We consider: regular solutions. We are interested in the limit. We take into account: general initial data thermal conduction general state laws. The analysis contains (at least) two parts: 1. an existence uniform boundedness result for a time interval independent of the small parameters. 2. a convergence result to the solution of a limit equation. We (essentially) restrict ourselves to: the whole space case. / Porquerolles France p.4/14

10 Many results available Five key assumptions: isentropic / nonisentropic; perfect gases / general state laws; inviscid / viscous fluid; unbounded / bounded domains; prepared / general initial data; / Porquerolles France p.5/14

11 Many results available Five key assumptions: isentropic / nonisentropic; perfect gases / general state laws; inviscid / viscous fluid; unbounded / bounded domains; prepared / general initial data; Existence of the solutions for a time independent of : S. Klainerman A. Majda (isentropic inviscid or vicous fluids general data or ). S. Schochet (nonisentropic Euler bounded domains prepared data). G. Métivier S. Schochet (nonisentropic Euler general data or ). R. Danchin (isentropic N.S. or global solutions with critical regularity small data) T. A. (nonisentropic Euler general data bounded or exterior domains). / Porquerolles France p.5/14

12 Many results available Five key assumptions: isentropic / nonisentropic; perfect gases / general state laws; inviscid / viscous fluid; unbounded / bounded domains; prepared / general initial data; Existence of the solutions for a time independent of : S. Klainerman A. Majda (isentropic inviscid or vicous fluids general data or ). S. Schochet (nonisentropic Euler bounded domains prepared data). G. Métivier S. Schochet (nonisentropic Euler general data or ). R. Danchin (isentropic N.S. or global solutions with critical regularity small data) T. A. (nonisentropic Euler general data bounded or exterior domains). Incompressible limit S. Klainerman A. Majda S. Schochet for prepared data. S. Ukai (isentropic Euler general initial data whole space). H. Isozaki (isentropic Euler general data exterior domain). G. Métivier S. Schochet (results in advances in ). T. A. (exterior domains). / Porquerolles France p.5/14

13 Many results available Five key assumptions: isentropic / nonisentropic; perfect gases / general state laws; inviscid / viscous fluid; unbounded / bounded domains; prepared / general initial data; Existence of the solutions for a time independent of : S. Klainerman A. Majda (isentropic inviscid or vicous fluids general data or ). S. Schochet (nonisentropic Euler bounded domains prepared data). G. Métivier S. Schochet (nonisentropic Euler general data or ). R. Danchin (isentropic N.S. or global solutions with critical regularity small data) T. A. (nonisentropic Euler general data bounded or exterior domains). Incompressible limit S. Klainerman A. Majda S. Schochet for prepared data. S. Ukai (isentropic Euler general initial data whole space). H. Isozaki (isentropic Euler general data exterior domain). G. Métivier S. Schochet (results in advances in ). T. A. (exterior domains). Many other results. Among others: D. Bresch R. Danchin B. Desjardins I. Gallagher E. Grenier D. Hoff C.K. Lin P.L. Lions N. Masmoudi S. Schochet.... / Porquerolles France p.5/14

14 Uniform stability Theorem (T. A. 04) Uniform existence of solutions having general initial data in appropriate Sobolev spaces together with uniform estimates. / Porquerolles France p.6/14

15 Uniform stability there in the Sobolev space Theorem (T. A. 04) Let be an integer large enough. For all positive is a positive such that for all all initial data take positive values such that the Cauchy problem has a unique solution such that depending only on take positive values. In addition there is a positive such that / Porquerolles France p.6/14

16 Uniform stability Theorem (T. A. 04) Let be an integer large enough. For all positive is a positive such that for all all initial data such that take positive values there in the Sobolev space the Cauchy problem has a unique solution such that take positive values. In addition there is a positive such that depending only on Remark : one can replace by with. In addition for perfect gases one can replace by or with. / Porquerolles France p.6/14

17 Singular system : Step of the proof defined by The triple solves (ANS) where. The coefficients satisfy / Porquerolles France p.7/14

18 Important Features of the System 1) Instability: The linearized equations are not uniformly wellposed. The reason is or 2) No estimates available when The reason is that we cannot factorize / Porquerolles France p.8/14

19 Step of the proof : the norm. DEFINITION Let with where / Porquerolles France p.9/14

20 Step of the proof : the norm. DEFINITION Let with where / Porquerolles France p.9/14

21 THEOREM Let be an integer. For all integer for all real there is a positive a positive such that for all all initial data in the Cauchy problem has a unique classical solution in. / Porquerolles France p.10/14

22 THEOREM Let be an integer. For all integer for all real there is a positive a positive such that for all all initial data in the Cauchy problem has a unique classical solution in. Estimates for the linearized system (additional smoothing effect); Estimate from the slow components: ; Estimate for the incompressible component ( High frequencies estimates : the commutators between operators can be seen as source terms); ); the singular / Porquerolles France p.10/14

23 THEOREM Let be an integer. For all integer for all real there is a positive a positive such that for all all initial data in the Cauchy problem has a unique classical solution in. Estimates for the linearized system (additional smoothing effect); Estimate from the slow components: ; Estimate for the incompressible component ( High frequencies estimates : the commutators between operators can be seen as source terms); ); the singular The main part: Low frequencies estimates Firstly estimates for. Secondly induction using the equations. / Porquerolles France p.10/14

24 The limit system. solves Fix The triple is the solution of where We expect that / Porquerolles France p.11/14

25 The limit system. solves Fix The triple is the solution of where We expect that with appropriate initial data. / Porquerolles France p.11/14 References : P. Embid R. Danchin S. Dellacherie P.L.Lions...?

26 Decay to of the local energy Only partial result PROPOSITION (T. A.) Fix. Assume that satisfy (ANS) for some fixed large enough. Suppose that the initial data converge in to a function decaying sufficiently rapidly at infinity in the sense that for some positive constants.. Then converge strongly to in / Porquerolles France p.12/14

27 Proof The proof relies upon a Theorem of G. Métivier S. Schochet : If where is bounded in is bounded in decay to zero at spatial infinity in to.. Then we multiply the momentum equation by to Then converges strongly to We apply this rersult with deduce convergence for. / Porquerolles France p.13/14

28 Conclusion Considering regular solutions defined in the whole space taking into account: general initial data thermal conduction general state laws an existence uniform boundedness result for a time interval independent of the small parameters has been stated. / Porquerolles France p.14/14

29 Conclusion Considering regular solutions defined in the whole space taking into account: general initial data thermal conduction general state laws an existence uniform boundedness result for a time interval independent of the small parameters has been stated. Concerning the convergence it has been stated that: / Porquerolles France p.14/14

30 Conclusion Considering regular solutions defined in the whole space taking into account: general initial data thermal conduction general state laws an existence uniform boundedness result for a time interval independent of the small parameters has been stated. Concerning the convergence it has been stated that: Additional results for perfect gases. / Porquerolles France p.14/14