Well-posedness of modified Camassa–Holm equations

Well-posedness of modified Camassa–Holm equations

Article history: Received 4 April 2008 Revised 11 January 2009 Available online 28 February 2009

Well-posedness of the Water-waves Equations

1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function ζ(t,X), where t denotes th...

Well-posedness for the Navier-Stokes equations

where u is the velocity and p is the pressure, with inital data u(x, 0) = u0(x). Existence of weak solutions has been shown by Leray. Uniqueness (and regularity) of weak solutions is unknown and both are among the major open questions in applied analysis. Under stronger assumptions there exist local and/or global smooth solutions. One version of this has been shown by Kato for initial data in L...

The well - posedness of Cauchy problem for dissipative modified Korteweg de Vries equations ∗

Abstract. In this paper we consider some dissipative versions of the modified Korteweg de Vries equation ut+uxxx+ |Dx| u+uux = 0 with 0 < α ≤ 3. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s > 1/4−α/4 on the basis of the [k; Z]−multiplier norm estimate obtained by Tao in [9] for KdV equation. 2000 Mathematics Subject Classification: 35Q5...

WELL- POSEDNESS OF THE ROTHE DIFFERENCE SCHEME FOR REVERSE PARABOLIC EQUATIONS