Well-posedness for regularized nonlinear dispersive wave equations

In this essay, we study the initial-value problem u(t) + u(x) + g(u)(x) + Lu(t) = 0, x is an element of R, t > 0, u(x,0) = u(0)(x), x is an element of R} (0.1) where u=u(x,t) is a real-valued function, L is a Fourier multiplier operator with real symbol alpha(xi), 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 L(p), H(s) and bore-like or kink-like initial data. In the special case where alpha(xi) = vertical bar xi vertical bar(r) wherein r > 1 and g(u) = 1/2 u(2), (0.1) is globally well-posed in time if s and r satisfy a simple algebraic relation.