Well-posedness for regularized nonlinear dispersive wave equations
In this essay, we study the initial-value problem (Equation Presented) where u = u(x, t) is a real-valued function, L is a Fourier multiplier operator with real symbol α(ξ), say, and g is a, smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for Lp, Hs and bore-like or kink-like initial data. In the special case where α(ξ) = |ξ| r wherein r > 1 and g(u) = 1/2u2, (0.1) is globally well-posed in time if s and r satisfy a simple algebraic relation.
Discrete and Continuous Dynamical Systems
Bona, J., & Chen, H. (2009). Well-posedness for regularized nonlinear dispersive wave equations. Discrete and Continuous Dynamical Systems, 23 (4), 1253-1275. https://doi.org/10.3934/dcds.2009.23.1253