Nonresonance and Global Existence of Prestressed Nonlinear Elastic Waves

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 configuration, 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 finite 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 difference roughly, our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to first order along the characteristic cones. Here, only the cancellation of nonlinear wave interactions among individual wave families is required. The difficulty 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.