<p>In this paper, we establish the global existence for incompressible Hookean elastodynamics exterior to star-shaped regions in three space dimensions, by using the invariance of the corresponding system under translations, simultaneous rotations, scaling and generalized energy estimates, provided that the displacement and pressure are null on the boundary and the initial data are small in some sense. To this end, we face two main difficulties, the first of which comes from the estimate of the pressure: the derivative loss for the higher-order estimate, which is overcome by using <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sum\nolimits_{\vert a\vert \leqslant N}\Vert\square\tilde{\Gamma}^{a}u\Vert_{L^{2}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mo movablelimits="false">∑</mo> <mrow> <mo fence="false" stretchy="false">∣</mo> <mi>a</mi> <mo fence="false" stretchy="false">∣</mo> <mo>⩽</mo> <mi>N</mi> </mrow> </msub> <mo>∥</mo> <mi>◻</mi> <msup> <mrow> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> <mrow> <mi>a</mi> </mrow> </msup> <mi>u</mi> <msub> <mo fence="false" stretchy="false">∥</mo> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> to absorb the nonlinear terms containing the highest-order derivatives of displacement (see Theorem 3.7), while the low-order estimate of the pressure is done by exploiting the quadratic nonlinear terms delicately to obtain better decay (see Lemma 3.9). The second one is caused by the boundary, compared with the corresponding Cauchy problem, since the angular momentum and the scaling operators do not preserve the null boundary condition. We overcome this difficulty by using the modified vector fields and the space-time (Keel-Smith-Sogge) estimate for the wave equation in the exterior domain (see Theorem 2.2) to control the local terms.</p>

Global existence of incompressible Hookean elastodynamics exterior to star-shaped regions in three dimensions

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发表日期：

2026年1月4日

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Wei Xu,
Ning-An Lai

摘要

In this paper, we establish the global existence for incompressible Hookean elastodynamics exterior to star-shaped regions in three space dimensions, by using the invariance of the corresponding system under translations, simultaneous rotations, scaling and generalized energy estimates, provided that the displacement and pressure are null on the boundary and the initial data are small in some sense. To this end, we face two main difficulties, the first of which comes from the estimate of the pressure: the derivative loss for the higher-order estimate, which is overcome by using $\sum\nolimits_{\vert a\vert \leqslant N}\Vert\square\tilde{\Gamma}^{a}u\Vert_{L^{2}}$ $\sum_{∣ a ∣ ⩽ N} ∥ ◻ {\tilde{Γ}}^{a} u ∥_{L^{2}}$ to absorb the nonlinear terms containing the highest-order derivatives of displacement (see Theorem 3.7), while the low-order estimate of the pressure is done by exploiting the quadratic nonlinear terms delicately to obtain better decay (see Lemma 3.9). The second one is caused by the boundary, compared with the corresponding Cauchy problem, since the angular momentum and the scaling operators do not preserve the null boundary condition. We overcome this difficulty by using the modified vector fields and the space-time (Keel-Smith-Sogge) estimate for the wave equation in the exterior domain (see Theorem 2.2) to control the local terms.