Well-posedness of the hydrostatic Navier–Stokes equations

On the Local Well-posedness of the Prandtl and Hydrostatic Euler Equations with Multiple Monotonicity Regions

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum u0 is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold....

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...

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 ROTHE DIFFERENCE SCHEME FOR REVERSE PARABOLIC EQUATIONS