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Local well-posedness for incompressible neo-Hookean elastic equations in almost critical Sobolev spaces

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Abstract

Inspired by a pioneer work of Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023), we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to \(H^{\frac{n+2}{2}+}({\mathbb {R}}^n) \times H^{\frac{n}{2}+}({\mathbb {R}}^n)\) (\(n=2,3\)), where \(\frac{n+2}{2}\) and \(\frac{n}{2}\) is respectively a scaling-invariant exponent for deformation and velocity in Sobolev spaces. Our new observation relies on two folds: a reduction to a second-order wave-elliptic system of deformation and velocity; and a “wave-map type” null form intrinsic in this coupled system. In particular, the wave nature with “wave-map type” null form allows us to prove a bilinear estimate of Klainerman–Machedon type for nonlinear terms. So we can lower \(\frac{1}{2}\)-order regularity in 3D and \(\frac{3}{4}\)-order regularity in 2D for well-posedness compared with Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023).

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Notes

  1. In the summation the repeated indices run from 1 to n.

  2. Our result also works for \(\textbf{x}=A\textbf{y}+\textbf{U}(t,\textbf{y})\), where A is a constant matrix.

  3. In our paper, we also call \(Q_0\)-type as wave-map type.

  4. The wave maps has \(Q_0\)-type null form, which is also a special model of (1.23).

  5. The space \({\mathcal {S}}'(\mathbb {R}^{1+n})\) is the dual space of Schwartz functions.

  6. Here (2.12) is equivalent to \(\text {det}\textbf{F}=1\).

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Acknowledgements

The authors sincerely express a great attitude to the reviewers for their helpful comments. The author is supported by National Natural Science Foundation of China (Grant No. 12101079) and the Fundamental Research Funds for the Central Universities (Grant No. 531118010867).

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  1. School of Mathematics, Hunan University, Lushan South Road in Yuelu District, Changsha, 410882, People’s Republic of China

    Huali Zhang

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Zhang, H. Local well-posedness for incompressible neo-Hookean elastic equations in almost critical Sobolev spaces. Calc. Var. 63, 66 (2024). https://doi.org/10.1007/s00526-024-02681-0

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