Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws
Tóm tắt
The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs’ regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin–Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta–Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.
Tài liệu tham khảo
Aregba-driollet D., Natalini R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000)
Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, 2011
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)
Chae D.: On the system of conservation laws and its perturbation in the Besov spaces. Adv. Differ. Eqs. 10, 983–1006 (2005)
Chae D.: On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002)
Chemin J. Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. Journal d’Analyse Mathématique 77, 27–50 (1999)
Chemin, J. Y.: Localization in Fourier space and Navier–Stokes system. In: Phase Space Analysis of Partial Differential Equations, Proceedings, CRM Series, pp. 53–136, 2004
Chen G.-Q., Levermore C.D., Liu T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)
Dafermos, C.M.: Can dissipation prevent the breaking of waves? In: Transactions of the Twenty-Sixth Conference of Army Mathematicians, pp. 187–198, ARO Rep. 81, 1, U. S. Army Res. Office, Research Triangle Park, N.C., 1981
Dafermos C.M.: Hyperbolic conservation laws in continuum physics, 3rd edn. Springer, Berlin, 2010
Danchin , Danchin : Local theory in the critical spaces for compressible viscous and heat-conductive gases. Commun. Part. Differ. Equs. 26, 1183–1233 (2001)
Evans, L.C.: Partial Differential Equations. Amer Mathematical Society, Rhode Island, 1998
Friedrichs K.O., Lax P.D.: Systems of conservation equations with a convex exttension. Proc. Natl. Acad. Sci. USA 68, 1686–1688 (1971)
Fang, D.Y., Xu, J.: Existence and asymptotic behavior of \({\mathcal{C}^1}\) solutions to the multidimensional compressible Euler equations with damping. Nonlinear Anal. TMA 70, 244–261 (2009)
Godunov S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR 139, 521–523 (1961)
Hsiao L.: Quasilinear Hyperbolic Systems and Dissipative Mechanisms. World Scientific Publishing, Singapore, 1997
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)
Huang F., Pan R.: Convergence rate for compressible Euler equations with damping and vacuum. Arch. Rational Mech. Anal. 166, 359–376 (2003)
Huang F., Marcati P., Pan R.: Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Rational Mech. Anal. 176, 1–24 (2005)
Iftimie D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matemática Iberoamericana 15, 1–36 (1999)
Iguchi T., Kawashima S.: On space-time decay properties of solutions to hyperbolic-elliptic coupled systems, Hiroshima Math. J. 32, 229–308 (2002)
Kato T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205 (1975)
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
Kawashima S., Yong W.-A.: Dissipative structure and entropy for hyperbolic systems of balance laws. Arch. Rational Mech. Anal. 174, 345–364 (2004)
Kawashima S., Yong W.-A.: Decay estimates for hyperbolic balance laws. J. Anal. Appl. 28, 1–33 (2009)
Jin S., Xin Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995)
Majda A.: Compressible Fluid Flow and Conservation laws in Several Space Variables. Springer, Berlin (1984)
Matsumura A.: An energy method for the equations of motion of compressible viscous and heat-conductive fluids, MRC Technical Summary Report, Univ. of Wisconsin-Masison, \({\sharp}\) 2194, 1981
Matsumura A., Nishida T.: The initial value problem for the quations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Nishida, T.: Nonlinear hyperbolic equations and relates topics in fluid dynamics. Publ. Math. D’Orsay, 46–53 (1978)
Ruggeri T., Serre D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Q. Appl. Math. 62, 163–179 (2004)
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)
Sideris T., Thomases B., Wang D.H.: Long time behavior of solutions to the 3D compressible Euler with damping. Commun. Part. Differ. Equ. 28, 953–978 (2003)
Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
Wang W., Yang T.: The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. J. Differ. Eqs. 173, 410–450 (2001)
Xu J.: Relaxation-time limit in the isothermal hydrodynamic model for semiconductors. SIAM J. Math. Anal. 40, 1979–1991 (2009)
Xu J., Xiong J., Kawashima S.: Global well-posedness in critical Besov spaces for two-fluid Euler–Maxwell equations. SIAM J. Math. Anal. 45, 1422–1447 (2013)
Yong W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal. 172, 247–266 (2004)
Zeng Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rational Mech. Anal. 150, 225–279 (2004)