zbMATH — the first resource for mathematics

The optimal decay estimates on the framework of Besov spaces for generally dissipative systems. (English) Zbl 1323.35141
Summary: We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic-parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the \({L^{p}(\mathbb{R}^{n})}\) embedding and the improved Gagliardo-Nirenberg inequality, the optimal \({L^{p}(\mathbb{R}^{n})-L^{2}(\mathbb{R}^{n})(1\leqq p < 2)}\) decay rates and \({L^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})(1\leqq p < 2\leqq q\leqq \infty)}\) decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.

35Q31 Euler equations
42B25 Maximal functions, Littlewood-Paley theory
76N15 Gas dynamics, general
Full Text: DOI arXiv
[1] Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin/Heidelberg, 2011 · Zbl 1227.35004
[2] Bianchini, S.; Hanouzet, B.; Natalini, R., Asymptotic behavior of smooth solutions for partially dissipative hyperoblic systems with a convex entropy, Commun. Pure Appl. Math., 60, 1559-1622, (2007) · Zbl 1152.35009
[3] Chemin, J.-Y.; Lerner, N., Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differ. Equ., 121, 314-328, (1995) · Zbl 0878.35089
[4] Chen, G.-Q.; Levermore, C.D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47, 787-830, (1994) · Zbl 0806.35112
[5] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics (Third Editor). Springer, Berlin, 2010 · Zbl 1196.35001
[6] Friedrichs, K.O.; Lax, P.D., Systems of conservation equations with a convex exttension, Proc. Natl. Acad. Sci. USA, 68, 1686-1688, (1971) · Zbl 0229.35061
[7] Godunov, S.K., An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139, 521-523, (1961) · Zbl 0125.06002
[8] Gressman, P.T.; Strain, R.M., Global classical solutions of the Boltzmann equation without angular cut-off, J. Am. Math. Soc., 24, 771-847, (2011) · Zbl 1248.35140
[9] Hsiao, L.: Quasilinear Hyperbolic Systems and Dissipative Mechanisms. World Scientific Publishing, Singapore, 1997 · Zbl 0911.35003
[10] Hanouzet, B.; Natalini, R., Global existence of smooth solutions for partially disipative hyperbolic systems with a convex entropy. arch, Rational Mech. Anal., 169, 89-117, (2003) · Zbl 1037.35041
[11] Hoff, D.; Zumbrun, K., Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44, 603-676, (1995) · Zbl 0842.35076
[12] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58, 181-205, (1975) · Zbl 0343.35056
[13] Kawashima, S.: Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics. Doctoral Thesis, Kyoto University, 1984. http://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/97887 · Zbl 0587.35046
[14] Kawashima, S.; Yong, W.-A., Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 174, 345-364, (2004) · Zbl 1065.35187
[15] Kawashima, S.; Yong, W.-A., Decay estimates for hyperbolic balance laws, J. Anal. Appl., 28, 1-33, (2009) · Zbl 1173.35365
[16] Majda, A.: Compressible Fluid Flow and Conservation laws in Several Space Variables. Springer, Berlin, 1984 · Zbl 0537.76001
[17] Matsumura, A.: An Energy Method for the Equations of Motion of Compressible Viscous and Heat-Conductive Fluids. MRC Technical Summary Report, University of Wisconsin-Masison, \({♯}\)2194, 1981 · Zbl 1037.35041
[18] Nirenberg, L., On elliptic partial differential equation, Ann. Scu. Norm. Sup. Pisa, Ser. III, 13, 115-162, (1959) · Zbl 0088.07601
[19] Shizuta, Y.; Kawashima, S., Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14, 249-275, (1985) · Zbl 0587.35046
[20] Nishida, T., Nonlinear hyperbolic equations and relates topics in fluid dynamics, Publ. Math. D’Orsay, 46, 46-53, (1978)
[21] Ruggeri, T.; Serre, D., Stability of constant equilibrium state for dissipative balance laws system with a convex entropy, Quart. Appl. Math., 62, 163-179, (2004) · Zbl 1068.35067
[22] Sohinger, V., Strain, R.M.: The Boltzmann equation, Besov spaces, and optimal time decay rates in \({\mathbb{R}^{n}_{x}}\) . Adv. Math. 261, 274-332 (2014) · Zbl 1293.35195
[23] Sideris, T., Thomases, B., Wang, D.H.: Long time behavior of solutions to the 3D compressible Euler with damping. Communun. Part. Differ. Equ. 28, 953-978 (2003) · Zbl 1048.35051
[24] Tan Z., Wang G.C.: Large time behavior of solutions for compressible Euler equations with damping in \({\mathbb{R}^{3}}\) . J. Differ. Equ. 252, 1546-1561 (2012) · Zbl 1237.35131
[25] Ueda, Y.; Duan, R.-J.; Kawashima, S., Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Rational Mech. Anal., 205, 239-266, (2012) · Zbl 1273.35051
[26] Umeda, T.; Kawashima, S.; Shizuta, Y., On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1, 435-457, (1984) · Zbl 0634.76120
[27] Wu, J.H., Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263, 803-831, (2005) · Zbl 1104.35037
[28] Wang, W.; Yang, T., The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equ., 173, 410-450, (2001) · Zbl 0997.35039
[29] Xu, J.; Kawashima, S., Global classical solutions for partially dissipative hyperbolic system of balance laws. arch, Rational Mech. Anal., 211, 513-553, (2014) · Zbl 1293.35173
[30] Yong, W.-A.: Basic aspects of hyperbolic relaxation systems. Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Their Applications, Vol. 47 (Eds. Freistühler H. and Szepessy A.). Birkhäuser, Boston, 259-305, 2001 · Zbl 1017.35068
[31] Yong, W.-A., Entropy and global existence for hyperbolic balance laws, Arch. Rational Mech. Anal., 172, 247-266, (2004) · Zbl 1058.35162
[32] Zeng, Y., Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Rational Mech. Anal., 150, 225-279, (2004) · Zbl 0966.76079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.