Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition
Tóm tắt
In the context of hyperbolic systems of balance laws, the Shizuta–Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition (SK) in 3D, such that some eigendirections do not exhibit dissipation at all. We prove that if the source term is nonresonant (in a suitable sense) in the direction where dissipation does not play any role, then the formation of singularities is prevented, despite the lack of dissipation, and the smooth solutions exist globally in time. The main idea of the proof is to couple Green function estimates for weakly dissipative hyperbolic systems with the space–time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah. More precisely, the partially dissipative hyperbolic systems violating (SK) are endowed, in the nondissipative directions, with a special structure of the nonlinearity, the so-called nonresonant bilinear form for the wave equation (see Pusateri and Shatah, CPAM 2013).
Tài liệu tham khảo
K. Beauchard, E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal. 199 , no. 1 (2011), 177–227.
F. Bernicot, P. Germain, Oscillatory integrals and boundedness for new bilinear multipliers. Adv. Math. 225, no. 4 (2010), 1739–1785.
R. Bianchini, Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling, SIAM J. Math. Anal. 50 (2) (2018), 1877-1899.
S. Bianchini, B. Hanouzet, R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), 1559-1622.
R. Bianchini, G. Staffilani, Revisitation of a Tartar’s result on a semilinear hyperbolic system with null condition, Fluid Dynamics, Dispersive Perturbations and Quantum Fluids, Springer UMI Series (2021), in press.
Y. Brenier, L. Corrias, A kinetic formulation for multi-branch entropy solutions of scalar conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (2) (1998), 169–190.
Y. Brenier, L. Corrias, R. Natalini, Relaxation limits for a class of balance laws with kinetic formulation, Advances in Nonlinear Partial Differential Equations and Related Areas, Beijing, 1997, pp. 2–14, World Sci. Publ., River Edge, 1998.
T. Buckmaster, S. Shkoller, V. Vicol, Formation of Shocks for 2D Isentropic Compressible Euler, Comm. Pure Appl. Math. 2020, in press.
G. Carbou, B. Hanouzet, Comportement semi-linéaire d’un systeme hyperbolique quasi-linéaire: le modèle de Kerr-Debye (French) [Semilinear behavior for a quasilinear hyperbolic system: the Kerr-Debye model]. C. R. Math. Acad. Sci. Paris 343(4) (2006), 243–247.
G. Carbou, B. Hanouzet, R. Natalini, Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation. J. Differ. Equ. 246(1) (2009), 291–319.
G.Q Chen, C.D. Levermore, T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 7(6) (1994), 787–830.
D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure App. Math. 39 (1986), 267-282.
D. Christodoulou, The formation of shocks in 3-dimensional fluids. EMS Monographs in Mathematics. European Mathematical Society (EMS), 2007.
R. Coifman, Y. Meyer, Commutators of singular integrals, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. I, pp. 265–273, Colloq. Math. Soc. János Bolyai, 19, North-Holland, Amsterdam-New York, 1978. MR0540305.
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin 1999.
E. V. Ferapontov, J. Moss, Linearly degenerate partial differential equations and quadratic line complexes, Comm. Anal. Geom. 23(1) (2015), 91-127.
V. Georgiev, S. Lucente, G. Ziliotti, Decay estimates for hyperbolic systems, Hokkaido Math. J. 33 (2004), 83-113.
P. Germain, N. Masmoudi, Global existence for the Euler-Maxwell system, Annales Scientifiques de l’École Normale Supérieure 47 (3) (2014), 469-503.
P. Germain, N. Masmoudi, J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN 3 (2009), 414-432. MR2482120.
P. Germain, N. Masmoudi, J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. 175 (2012), 691-754.
P. Germain, Space-time resonances, Exp. 8, Proceedings of the Journées EDP 2010.
B. Hanouzet, J.-L. Joly, Applications bilinéaires compatibles avec un opérateur hyperbolique, Ann. Inst. H. Poincaré Anal. Non Lineaire 4 (1987), 357-376.
B. Hanouzet, R. Natalini, Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy, Arch. Rational Mech. Anal. 169 (2003), 89-117.
K. Ide, S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008), no. 7, 1001–1025
S. Jin, Z. Xin, The relaxation schemes for system of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235-277.
F. John, Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29-51.
F. John, Nonlinear Wave Equations, Formation of Singularities, Pitcher Lectures in Math. Sciences, Leigh University, Amer. Math. Soc., 1990.
D. Lannes, Séminaire BOURBAKI, 64ème année, 2011-2012, no 1053.
T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow. J. Differ. Equ. 190 (2003), 131–149.
T. Kato, Perturbation theory for linear operator, 2nd ed. Grundlehren der Mathematischen Wissenschaften, 132. Springer, New York (1976).
S. Klainerman, Global Existence for Nonlinear Wave Equations, Comm. Pure Appl. Math. 23 (1980), 43-101.
S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), volume 23 of Lectures in Appl. Math., pages 293- 326. Amer. Math. Soc., Providence, RI, 1986.
A. Majda, Compressible fluid flow and systems of conservation laws in several space dimensions, Appl. Math. Sci., vol. 53, Springer, New York 1978.
C. Mascia, R. Natalini, On relaxation hyperbolic systems violating the Shizuta-Kawashima condition. Arch. Ration. Mech. Anal. 195 (2010), no. 3, 729–762.
G. Métivier, Para-differential calculus and applications to the Cauchy problem for nonlinear systems. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, 5. Edizioni della Normale, Pisa, 2008. xii+140 pp.
T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, 1978, Publications Mathématiques d’Orsay, No. 78-02.
F. Pusateri, J. Shatah, Space-Time Resonances and the Null Condition for First-Order Systems of Wave Equations, Comm. Pure and Appl. Math. 66 (2013), 1495-1540.
J. Shatah, Normal Forms and Quadratic Nonlinear Klein-Gordon Equations, Comm. Pure and Appl. Math. 38 (1985), 685-696.
Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) n.2, 249-275.
T.C. Sideris, B. Thomases, D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping. Commun. Partial Differ. Equ. 28(3–4) (2003), 795–816.
T. C. Sideris, B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics, Communications on Pure and Applied Mathematics, 60 (2007), 1707–1730.
J. Luk, J. Speck, The hidden null structure of the compressible Euler equations and a prelude to applications. J. Hyperbolic Differ. Equ. 17 (2020), n.1, 1-60.
C. Wei, Y.-Z. Wang, Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases. J. Hyperbolic Differ. Equ. 17 (2020), n.3, 613-637.
T. Tao, Lecture notes 6 for 247B1 (Paraproducts), https://www.math.ucla.edu/ tao/247b.1.07w/notes6.pdf.
L. Tartar, Compensated compactness and applications to partial differential equations, Non linear analysis and mechanics: Heriot Watt Symposium, vol. III, Research Notes in Mathematics, Pitman.
L. Tartar, Some existence theorems for semilinear hyperbolic systems in one space variable (1981), unpublished.
G.B. Whitham, Linear and nonlinear waves. Wiley & Sons, New York, 1974.
W.-A. Yong, Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266.
Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Rational. Mech. Anal. 150 (1999), 225–279.
R.-W. Ziolkowski, The incorporation of microscopic material models into FDTD approach for ultrafast optical pulses simulations, IEEE Transactions on Antennas and Propagation 45 (1997), no. 3, 375–391.