We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is measured.1991 Mathematics Subject Classification. Primary: 35Q53. Secondary: 35B05.

In this article, we examine L 2 well-posedness and stabilization property of the dispersion-generalized Benjamin-Ono equation with periodic boundary conditions. The main ingredient of our proof is a development of dissipation-normalized Bourgain space, which gains smoothing properties simultaneously from dissipation and dispersion within the equation. We will establish a bilinear estimate for the derivative nonlinearity using this space and prove the linear observability inequality leading to small-data stabilization.

Abstract. This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equationThe main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times. IntroductionIn this work we study propagation of regularity and persistence of decay results for a class of fifth order dispersive models. For concreteness, the main theorems are stated for initial value problems of the formwhere c j are real constants, u : R × R → R is an unknown function and u 0 : R → R is a given function. Eq. (1.1) contains the specific equationwhich is the third equation in the sequence of nonlinear dispersive equationsknown as the KdV hierarchy. Here the polynomials Q j are chosen so that equation (1.3) has the Lax pair formulationThe first two equations in the hierarchy areand the KdV equationWith only slight modifications concerning the hypothesis on the initial data, the techniques in this paper apply to a large class of fifth order equations including the following models arising from mathematical physics:

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in H s (T), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.2010 Mathematics Subject Classification. Primary: 35Q53, 93B05, 93D15.