We generalize the notions of Levitin-Polyak well-posedness to an equilibrium problem with both abstract and functional constraints. We introduce several types of generalized Levitin-Polyak well-posedness. Some metric characterizations and sufficient conditions for these types of wellposedness are obtained. Some relations among these types of well-posedness are also established under some suitab...

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...

We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the BMO norm of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we...

We prove local well-posedness of the initial-boundary value problem for the Korteweg-de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.

The quasigeostrophic model is a simpli ed geophysical uid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasigeostrophic equation whose well-posedness is well-known, there see...

We show well-posedness of the Cauchy problem for the delay equation u′(t) = Au(t) + Φut, with initial values in X × Lp(−h, 0; X); X a Banach space, A the generator of a C0-semigroup on X, h ∈ {1,∞}, 1 6 p < ∞. In the first result, Φ is defined by a Stieltjes integral, and p = 1. In the second result, Φ is a continuous linear mapping from W 1 p (−h, 0; X) to the Favard class of A. MSC 2000: 34K06

We prove local well-posedness results for the semi-linear wave equation for data in H , 0 < < n?3 2(n?1) , extending the previously known results for this problem. The improvement comes from an introduction of a two-scale Lebesgue space X r;p k .

Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as ‘flows’ on hypersurface geometries are considered. It is shown that these hypersurface ‘flows’ are not generally well-posed for smooth initial data and counterexamples to existence are constructed.