## The pointwise estimates of solutions for Euler equations with damping in multi-dimensions.(English)Zbl 0997.35039

The paper deals with the time-asymptotic behavior of solutions to the isentropic Euler equations with damping in several space dimensions. The main purpose is to study the pointwise estimates of solutions. The case when the solution is a perturbation of a steady state is considered. The analysis is based on some estimates to the solutions by using energy methods, and the estimates of the Green function to the linearized system about the steady state. It is proved that the time-asymptotic shape of the solution is the diffusion wave. The optimal $$L_p$$, $$1<p\leq\infty$$, convergence rate of the solution is obtained.

### MSC:

 35Q05 Euler-Poisson-Darboux equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B40 Asymptotic behavior of solutions to PDEs
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### References:

 [1] Dafermos, C., A system of hyperbolic conservation laws with frictional damping, Z. angew math. phys., 46, 294-307, (1995) · Zbl 0836.35091 [2] Evans, L.C., Partial differential equations, Graduate studies in math., 19, (1998), Amer. Math. Soc Providence [3] Hoff, D.; Zumbrun, K., Multi-dimensional diffusion wave for the navier – stokes equations of compressible flow, Indiana univ. math. journal, 44, 603-676, (1995) · Zbl 0842.35076 [4] Hoff, D.; Zumbrun, K., Pointwise decay estimates for multidimensional navier – stokes diffusion waves, Z. angew math. phys., 48, 1-18, (1997) · Zbl 0882.76074 [5] Hsiao, L.; Liu, T.-P., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. math. phys., 143, 599-605, (1992) · Zbl 0763.35058 [6] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. rational mech. anal., 58, 181-205, (1945) · Zbl 0343.35056 [7] Kawashima, S., System of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, (1983), Kyoto University Kyoto [8] Liu, T-P., Pointwise convergence to shock waves for viscous conservation laws, Commun. pure appl. math., L, 1113-1182, (1997) · Zbl 0902.35069 [9] T. P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Commun. Math. Phys.1961998, 145-173. · Zbl 0912.35122 [10] Liu, T.P.; Zeng, Y., Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. amer. math. soc., 599, (1997) [11] C. Lattanzio, and, P. Marcati, Asymptotic stability of plane diffusion waves for the 2-D quasilinear wave equation, submitted for publication. · Zbl 0945.35011 [12] Makino, T.; Ukai, S., Sur l’existence des solution locales de l’équation d’euler – poisson pour l’évolution gazeuses, J. math. Kyoto univ., 27, 387-399, (1987) · Zbl 0657.35113 [13] Nishida, T., Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications mathématiques D’orsay 78.02, (1978), Départment de Mathématique Paris-sud [14] Nishihara, K., Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. differential equations, 131, 171-188, (1996) · Zbl 0866.35066 [15] Nishihara, K.; Wang, W.; Yang, T., L^{p}-convergence rate to nonlinear diffusion waves for p-system with damping, J. differential equations, 161, 191-218, (2000) · Zbl 0946.35012 [16] H. Xu, and, W. Wang, Large time behavior of solutions for isentropic Navier-Stokes equations in even space dimension, Acta Math. Sci, in press.
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