Propagation of density-oscillations in solutions to the compressible Navier-Stokes-Poisson system

Abstract

Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system (HCNSP) which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to (CNSP) (see Corollary 6.1) are also obtained.

Keywords

Compressible fluids / Navier-Stokes-Poisson equations / Young measures / Propagation of oscillations / Strong solutions

References

[1] Agmon S., Douglis A., Nirenberg L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., 1964, 17: 35-92

[2] Ball, J. M., A version of the fundamental theorem for Young measures, Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre and M. Slemrod (eds.), Lecture Notes in Physics, 344, Springer-Verlag, 1989, 207–215.

[3] Cho Y., Choe H. J., Kim H.. Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl., 2004, 83: 243-275

[4] Feireisl E.. Dynamics of Viscous Compressible Fluids, 2004, Oxford: Oxford University Press

[5] Feireisl E., Novotný A., Petleltová H.. On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids. J. Math. Fluid Mech., 2001, 3: 358-392

[6] Hillairet M.. Propagation of density-oscillations in solutions to the barotropic Navier-Stokes system. J. Math. Fluid Mech., 2007, 9: 343-376

[7] Hoff D.. Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J., 1992, 41: 1225-1302

[8] Hungerbühler N.. Young Measures and Nonlinear PDEs, 2000, ETH Zürich: Habilitationsschrift

[9] Jiang S., Zhang P.. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commum. Math. Phys., 2001, 215: 559-581

[10] Kobayashi, T. and Suzuki, T., Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 2008, in press.

[11] Li T. T., Wang L. B.. Inverse piston problem for the system of one-dimensional isentropic flow. Chin. Ann. Math., 2007, 28B(3): 265-282

[12] Lions P. L.. Mathematical Topics in Fluids Mechanics, Vol. 2, Oxford Lecture Series in Mathematics and Its Applications, 1998, New York: The Clarendon Press University Press

[13] Novotný A., Straškraba I.. Introduction to the Mathematical Theory of Compressible Flow, 2004, New York: Oxford University Press

[14] Peng Y. J., Wang S.. Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations. Chin. Ann. Math., 2007, 28B(5): 583-602

[15] Serre D.. Variations de grande amplitude pour la densité d’un fluide visqueux compressible. Phys. D, 1991, 48: 113-128

[16] Vaigant V. A., Kazhikhov A. V.. On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (in Russian). Sibirskij Mat. Z., 1995, 36: 1283-1316

[17] Valadier M.. Cellina A.. Young measures, Methods of Nonconvex Analysis, 1990, Berlin: Springer-Verlag 152-188

[18] Valadier M.. A course on Young measures, Workshop on Measure Theory and Real Analysis (in Italian), Grado, 1993. Rend. Istit. Mat. Univ. Trieste, 1994, 26: 349-394

[19] Valli A.. Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann. Scuola. Norm. Sup. Pisa cl. sci., 1983, 10(4): 607-647

[20] Yin J. P., Tan Z.. Global existence of strong solutions of Navier-Stokes-Poisson equations for one-dimensional isentropic compressible fluids. Chin. Ann. Math., 2008, 29B(4): 441-458