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Scattering and rigidity for nonlinear elastic waves

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Abstract

For the Cauchy problem of nonlinear elastic wave equations of three-dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in Agemi (Invent Math 142:225–250, 2000) and Sideris (Ann Math 151:849–874, 2000), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument.

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Notes

  1. Note that the notation used for \(\Gamma ^a\) differs from the standard multi-index notation. If \(|a|=0\), \(\Gamma ^{a}\) just means the identity operator.

  2. Repeated indices are always summed.

  3. We truncate the nonlinearity at the quadratic level, because the higher order terms have no essential influence on the discussion of the global existence and asymptotic behavior of solutions with small amplitude.

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Acknowledgements

The author would like to express his sincere gratitude to the referee for his helpful comments and suggestions on this paper. The author is supported by National Natural Science Foundation of China (No.12371217) and the Fundamental Research Funds for the Central Universities (No.2232022D-27).

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  1. Department of Mathematics, Donghua University, Shanghai, 201620, People’s Republic of China

    Dongbing Zha

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Zha, D. Scattering and rigidity for nonlinear elastic waves. Calc. Var. 63, 115 (2024). https://doi.org/10.1007/s00526-024-02736-2

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