The 3D non-isentropic compressible Euler equations with damping in a bounded domain

Abstract

The authors investigate the global existence and asymptotic behavior of classical solutions to the 3D non-isentropic compressible Euler equations with damping on a bounded domain with slip boundary condition. The global existence and uniqueness of classical solutions are obtained when the initial data are near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

Keywords

Non-isentropic / Euler equations / Damping / Exponential convergence

References

[1] Bourguignon J. P., Brezis H.. Remarks on the Euler equation. J. Funct. Anal., 1975, 15: 341-363

[2] Dafermos C. M.. Can dissipation prevent the breaking of waves? Transactions of the Twenty-Sixth Conference of Army Mathematicians, 1981 187-198

[3] Duan R. J., Ma H. F.. Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. Indiana Univ. Math. J., 2008, 57(5): 2299-2319

[4] Duan R. J., Liu H. X., Ukai S., Yang T.. Optimal L ^p - L ^q convergence rates for the compressible Navier-Stokes equations with potential force. J. Differ. Equations, 2007, 238: 220-233

[5] Duan R. J., Ukai S., Yang T., Zhao H. J.. Optimal convergence rate for compressible Navier-Stokes equations with potential force. Math. Models Methods Appl. Sci., 2007, 17: 737-758

[6] Evans L. C.. Partial Differential Equations, 1998, Providence: Amer. Math. Soc.

[7] Fang D. Y., Xu J.. Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping. Nonlinear Anal., 2009, 70: 244-261

[8] Hsiao L.. Quasilinear Hyperbolic Systems and Dissipative Mechanisms, 1998, Singapore: World Scientific

[9] Hsiao L., Liu T. P.. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys., 1992, 143: 599-605

[10] Hsiao L., Liu T. P.. Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chin. Ann. Math. Ser. B, 1993, 14(1): 1-16

[11] Hsiao L., Luo T.. Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media. J. Differ. Equations, 1996, 125: 329-365

[12] Hsiao L., Pan R. H.. Initial-boundary value problem for the system of compressible adiabatic flow through porous media. J. Differ. Equations, 1999, 159: 280-305

[13] Hsiao L., Pan R. H.. The damped p-system with boundary effects. Contemporary Mathematics, 2000, 255: 109-123

[14] Hsiao L., Serre D.. Global existence of solutions for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal., 1996, 27: 70-77

[15] Huang F. M., Marcati P., Pan R. H.. Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal., 2005, 176: 1-24

[16] Huang F. M., Pan R. H.. Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum. J. Differ. Equations, 2006, 220: 207-233

[17] Huang F. M., Pan R. H.. Convergence rate for compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal., 2003, 166: 359-376

[18] Jiang M. N., Ruan L. Z., Zhang J.. Existence of global smooth solution to the initial boundary value problem for p-system with damping. Nonlinear Anal., 2009, 70(6): 2471-2479

[19] Jiang M. N., Zhang Y. H.. Existence and asymptotic behavior of global smooth solution for p-system with nonlinear damping and fixed boundary effect. Math. Meth. Appl. Sci., 2014, 37: 2585-2596

[20] Jiang M. N., Zhu C. J.. Convergence rates to nonlinear diffusion waves for p-system with nonlinear damping on quadrant. Discrete Contin. Dyn. Syst., 2009, 23(3): 887-918

[21] Jiang M. N., Zhu C. J.. Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant. J. Differ. Equations, 2009, 246(1): 50-77

[22] Liu T. P.. Compressible flow with damping and vacuum. Japan J. Appl. Math., 1996, 13: 25-32

[23] Marcati P., Milani A.. The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equations, 1990, 84(1): 129-147

[24] Marcati P., Rubino B.. Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differ. Equations, 2000, 162(2): 359-399

[25] Marcati P., Pan R. H.. On the diffusive profiles for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal., 2001, 33: 790-826

[26] Nishida T.. Global solutions for an initial-boundary value problem of a quasilinear hyperbolic systems. Proc. Japan Acad., 1968, 44: 642-646

[27] Nishida T.. Nonlinear Hyperbolic Equations and Relates Topics in Fluid Dynamics, 1978

[28] Pan R. H.. Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media. Michigan Math. J., 2001, 49: 519-539

[29] Pan R. H., Zhao K.. The 3D compressible Euler equations with damping in a bounded domain. J. Differ. Equations, 2009, 246: 581-596

[30] Schochet S.. The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. Comm. Math. Phys., 1986, 104: 49-75

[31] Sideris T. C., Thomases B., Wang D. H.. Long time behavior of solutions to the 3D compressible Euler equations with damping. Comm. Partial Differential Equations, 2003, 28: 795-816

[32] Tan Z., Wu G. C.. Large time behavior of solutions for compressible Euler equations with damping in R3. J. Differ. Equations, 2012, 252(2): 1546-1561

[33] Tan Z., Wang H. Q.. Global existence and optimal decay rate for the strong solutions in H2 to the 3-D compressible Navier-Stokes equations without heat conductivity. J. Math. Anal. Appl., 2012, 394(2): 571-580

[34] Wang W., Yang T.. The pointwise estimates of solutions for Euler equations with damping in multidimensions. J. Differ. Equations, 2001, 173: 410-450

[35] Wu G. C., Tan Z., Huang J.. Global existence and large time behavior for the system of compressible adiabatic flow through porous media in R3. J. Differ. Equations, 2013, 2553: 865-880

[36] Zhang Y. H., Tan Z.. Existence and asymptotic behavior of global smooth solution for p-system with damping and boundary effect. Nonlinear Anal., 2010, 72(5): 2499-2513

[37] Zhang Y. H., Wu G. C.. Global existence and asymptotic behavior for the 3d compressible nonisentropic Euler equations with damping. Acta Mathematica Sci. Ser. B, 2014, 34(5): 424-434

[38] Zhang Y. H., Zhu C. J.. Global existence and optimal convergence rates for the strong solutions in H2 to the 3D viscous liquid-gas two-phase flow model. J. Differ. Equations, 2015, 258(7): 2315-2338

[39] Zhao H. J.. Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J. Differ. Equations, 2001, 174: 200-236

[40] Zheng Y.. Global smooth solutions to the adiabatic gas dynamics system with dissipation terms. Chin. Ann. Math. Ser. A, 1996, 17: 155-162

[41] Zhu C. J.. Convergence rates to nonlinear diffusion waves for weak entropy solutions to p-system with damping. Sci. China, Ser. A, 2003, 46(4): 562-575

[42] Zhu C. J., Jiang M. N.. Lp-decay rates to nonlinear diffusion waves for p-system with nonlinear damping. Sci. China, Ser. A, 2006, 49(6): 721-739