Divergence-free positive symmetric tensors and fluid dynamics
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1209-1234.

We consider d×d tensors A(x) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of (det⁡A)1d−1. We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an a priori estimate for a new quantity, namely the space–time integral of ρ1np, where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

DOI : 10.1016/j.anihpc.2017.11.002
Mots-clés : Conservation laws, Gas dynamics, Functional inequalities 
@article{AIHPC_2018__35_5_1209_0,
     author = {Serre, Denis},
     title = {Divergence-free positive symmetric tensors and fluid dynamics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1209--1234},
     publisher = {Elsevier},
     volume = {35},
     number = {5},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.11.002},
     mrnumber = {3813963},
     zbl = {1393.35181},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2017.11.002/}
}
TY  - JOUR
AU  - Serre, Denis
TI  - Divergence-free positive symmetric tensors and fluid dynamics
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 1209
EP  - 1234
VL  - 35
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2017.11.002/
DO  - 10.1016/j.anihpc.2017.11.002
LA  - en
ID  - AIHPC_2018__35_5_1209_0
ER  - 
%0 Journal Article
%A Serre, Denis
%T Divergence-free positive symmetric tensors and fluid dynamics
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 1209-1234
%V 35
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2017.11.002/
%R 10.1016/j.anihpc.2017.11.002
%G en
%F AIHPC_2018__35_5_1209_0
Serre, Denis. Divergence-free positive symmetric tensors and fluid dynamics. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1209-1234. doi : 10.1016/j.anihpc.2017.11.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.11.002/

[1] Allaire, G. Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol. 146, Springer-Verlag, 2002 | DOI | MR | Zbl

[2] Anile, A.M. Relativistic Fluids and Magneto-Fluids, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 1989 | Zbl

[3] Ball, J.M. Convexity conditions and existence theorems in non-linear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337–403 | MR | Zbl

[4] Bony, J.-M. Journées EDPs (1987) (Saint Jean de Monts, 1987 Exp. # XVI) | MR

[5] Cercignani, C. Global weak solutions of the Boltzmann equation, J. Stat. Phys., Volume 118 (2005), pp. 333–342 | DOI | MR | Zbl

[6] Coifman, R.; Lions, P.-L.; Meyer, Y.; Semmes, S. Compacité par compensation et espaces de Hardy, C. R. Acad. Sci. Paris, Volume 309 (1989), pp. 945–949 | Numdam | MR | Zbl

[7] Conca, C.; Vanninathan, M. On uniform H2-estimates in periodic homogenization, Proc. R. Soc. Edinb. A, Volume 131 (2001), pp. 499–517 | DOI | MR | Zbl

[8] De Philippis, G.; Figalli, A. The Monge–Ampère equation and its link to optimal transportation, Bull., New Ser., Am. Math. Soc., Volume 51 (2014), pp. 527–581 | DOI | MR | Zbl

[9] DiPerna, R.; Lions, P.-L. On the Cauchy problem for the Boltzmann equation: global existence and weak stability results, Ann. Math., Volume 130 (1990), pp. 321–366 | MR | Zbl

[10] Federer, H. Geometric Measure Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1969 | MR | Zbl

[11] Figalli, A.; Maggi, F.; Pratelli, A. A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., Volume 182 (2010), pp. 167–211 | DOI | MR | Zbl

[12] Gagliardo, E. Proprietà di alcune di funzioni in più variabili, Ric. Mat., Volume 7 (1958), pp. 102–137 | MR | Zbl

[13] Gromov, M.; Milman, V.D.; Schechtman, G. Isoperimetric inequalities in Riemannian manifolds. Appendix I, Asymptotic Theory of Finite Dimensional Normed Space, Lect. Notes Math., vol. 1200, Springer-Verlag, 1986, pp. 114–129 | MR | Zbl

[14] Yan, Li Yan Some existence results of fully nonlinear elliptic equations of Monge–Ampère type, Commun. Pure Appl. Math., Volume 43 (1990), pp. 233–271 | MR | Zbl

[15] Lions, P.-L.; Masmoudi, N. From Boltzmann equation to the Navier–Stokes and Euler equations II, Arch. Ration. Mech. Anal., Volume 158 (2001), pp. 195–211 | MR | Zbl

[16] Makino, T.; Ukai, S. Local smooth solutions of the relativistic Euler equation, J. Math. Kyoto Univ., Volume 35 (1995), pp. 105–114 | MR | Zbl

[17] Müller, S. A surprising higher integrability property of mappings with positive determinant, Bull., New Ser., Am. Math. Soc., Volume 21 (1989), pp. 245–248 | DOI | MR | Zbl

[18] Murat, F. Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Volume 5 (1978), pp. 489–507 | Numdam | MR | Zbl

[19] Serre, D. Matrices, Grad. Texts Math., vol. 216, Springer-Verlag, 2002 | MR | Zbl

[20] Serre, D.; Friedlander, S.; Serre, D. Shock reflection in gas dynamics, Handbook of Mathematical Fluid Dynamics IV, Elsevier, North-Holland, 2007, pp. 39–122 | DOI | MR

[21] Serre, D. Expansion of a compressible gas in vacuum, Bull. Inst. Math. Acad. Sin. (N.S.), Volume 10 (2015), pp. 695–716 | MR | Zbl

[22] Tartar, L. Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. IV, Res. Notes in Math., vol. 39, Pitman, 1979, pp. 136–212 | MR | Zbl

[23] Tartar, L. The General Theory of Homogenization; A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, 2009 | MR | Zbl

[24] Villani, C. Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, Amer. Math. Society, 2003 | MR | Zbl

Cité par Sources :