Nonresonance and global existence of prestressed nonlinear elastic waves

By T. Sideris. 2000

202 citations recorded by the source corpus.

This article considers the existence of global classical solutions to the Cauchy problem in nonlinear elastodynamics. The unbounded elastic medium is assumed to be homogeneous, isotropic, and hyperelastic. As in the theory of 3D nonlinear wave equations in three space dimensions, global existence hinges on two basic assumptions. First, the initial deformation must be a small displacement from equilibrium, in this case a prestressed homogeneous dilation of the reference conflguration, and equally important, the nonlinear terms must obey a type of nonresonance or null condition. The omission of either of these assumptions can lead to the breakdown of solutions in flnite time. In particular, nonresonance complements the genuine nonlinearity condition of F. John, under which arbitrarily small spherically symmetric displacements develop singularities (although one expects this to carry over to the nonsymmetric case, as well), [4]. John also showed that small solutions exist almost globally [5] (see also [10]). Formation of singularities for large displacements was illustrated by Tahvildar-Zadeh [16]. The nonresonance condition introduced here represents a substantial improvement over our previous work on this topic [13]. To explain the difierence roughly, our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones. Here, only the cancellation of nonlinear wave interactions among individual wave families is required. The di‐culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal Helmholtz projection, which is ill-suited to nonlinear analysis. However, our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous, and this permits the usage of approximate local decompositions.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/802596868b7bd53f50b7892609b50dff6720f379

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Almost global existence for some semilinear wave equations (2001, background): “Further techniques and results concerning equations from the theory of elasticity were presented in [9], [10].”
Global existence of null-form wave equations in exterior domains (2006, methodology): “Introduction Inspired by the approach of Sideris and Tu [28] in the boundaryless case and the application of such techniques in Sideris [26], we prove global existence for multi-speed systems of quasilinear wave equations satisfying the null condition in certain exterior domains without using estima”