On the convergence of the nonlocal nonlinear model to the classical elasticity equation

Abstract

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional propagation of nonlinear waves in a continuous medium. In the limit of vanishing nonlocality we study the behavior of solutions to the Cauchy problem. We prove that, as the kernel functions of the convolution integral approach the Dirac delta function, the solutions converge strongly to the corresponding solutions of the classical elasticity equation. An energy estimate with no loss of derivative plays a critical role in proving the convergence result. As a typical example, we consider the continuous limit of the discrete lattice dynamic model (the Fermi-Pasta-Ulam-Tsingou model) and show that, as the lattice spacing approaches zero, solutions to the discrete lattice equation converge to the corresponding solutions of the classical elasticity equation.

arXiv comment17 pages

References in corpus (4)

• Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations
• The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials
• A conditional well-posedness result for the bidirectional Whitham equation
• Comparison of nonlocal nonlinear wave equations in the long-wave limit