Remarks on nonlinear elastic waves with null conditions

Zha, Dongbing; Peng, Weimin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 121.

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For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

[1] R. Agemi, Global existence of nonlinear elastic waves Invent. Math. 142 (2000) 225–250. | DOI | MR | Zbl

[2] A.J. Chorin, J.E. Marsden, A mathematical introduction to fluid mechanics, Texts in Applied Mathematics, 3rd edn. Vol. 4, Springer-Verlag, New York (1993). | DOI | MR | Zbl

[3] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39 (1986) 267–282. | DOI | MR | Zbl

[4] P.G. Ciarlet, Mathematical elasticity, Three-dimensional elasticity, Studies in Mathematics and its Applications, Vol. 20, North-Holland Publishing Co., Amsterdam (1988). | Zbl

[5] M.E. Gurtin, Topics in finite elasticity, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 35, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1981). | MR | Zbl

[6] K. Hidano and D. Zha, Space-time $L^2$ estimates, regularity and almost global existence for elastic waves. Forum Math. 30 (2018) 1291–1307. | DOI | MR | Zbl

[7] F. John, Blow-up for quasilinear wave equations in three space dimensions. Comm. Pure Appl. Math. 34 (1981) 29–51. | DOI | MR | Zbl

[8] F. John, Formation of singularities in elastic waves, in Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), Lecture Notes in Physics, Vol. 195, Springer, Berlin (1984) 194–210. | MR | Zbl

[9] F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances. Comm. Pure Appl. Math. 41 (1988) 615–666. | DOI | MR | Zbl

[10] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions. Comm. Pure Appl. Math. 37 (1984) 443–455. | DOI | MR | Zbl

[11] S. Katayama, Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. J. Differ. Equ. 255 (2013) 120–150. | DOI | MR | Zbl

[12] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., Vol. 23. American Mathematics Society, Providence, RI (1986) 293–326. | MR | Zbl

[13] S. Klainerman, On the work and legacy of Fritz John, 1934–1991. Comm. Pure Appl. Math. 51 (1998) 991–1017. | DOI | MR | Zbl

[14] S. Klainerman and T.C. Sideris, On almost global existence for nonrelativistic wave equations in 3D. Comm. Pure Appl. Math. 49 (1996) 307–321. | DOI | MR | Zbl

[15] H. Kubo, Lower bounds for the lifespan of solutions to nonlinear wave equations in elasticity, in Evolution equations of hyperbolic and Schrödinger type, Progress in Mathematics, Vol. 301. Birkhäuser/Springer Basel AG, Basel (2012) 187–212. | MR | Zbl

[16] T. Li and T. Qin, Physics and Partial Differential Equations. Volume 1. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Higher Education Press, Beijing (2012). | MR | Zbl

[17] T. Li, X. Yu, Life-span of classical solutions to fully nonlinear wave equations. Comm. Part. Differ. Equ. 16 (1991) 909–940. | DOI | MR | Zbl

[18] T.C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions. Comm. Part. Differ. Equ. 8 (1983) 1291–1323. | DOI | MR | Zbl

[19] T.C. Sideris, The null condition and global existence of nonlinear elastic waves. Invent. Math. 123 (1996) 323–342. | DOI | MR | Zbl

[20] T.C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. Math. 151 (2000) 849–874. | DOI | MR | Zbl

[21] T.C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal. 33 (2001) 477–488. | DOI | MR | Zbl

[22] J. Xin and T. Qin, Almost global existence for the initial value problem of nonlinear elastodynamic system. J. Math. Anal. Appl. 339 (2008) 517–529. | DOI | MR | Zbl

[23] D. Zha, Remarks on nonlinear elastic waves in the radial symmetry in $2-D$. Discrete Contin. Dyn. Syst. 36 (2016) 4051–4062. | DOI | MR | Zbl

[24] D. Zha, Space–time $L^2$ estimates for elastic waves and applications. J. Differ. Equ. 263 (2017) 1947–1965. | DOI | MR | Zbl

[25] D. Zha, On nonlinear elastic waves in $2-D$. J. Differ. Equ. 268 (2020) 1250–1269. | DOI | MR | Zbl

[26] D. Zha and K. Hidano, Global solutions to systems of quasilinear wave equations with low regularity data and applications. J. Math. Pures. Appl. 142 (2020) 146–183. | DOI | MR | Zbl