where := −∂2 t +∆ and u[0] := (u, ut)|t=0. The equation is semi-linear if F is a function only of u, (i.e. F = F (u)), and quasi-linear if F is also a function of the derivatives of u (i.e. F = F (u,Du), where D := (∂t,∇)). The goal is to use energy methods to prove local well-posedness for quasilinear equations with data (f, g) ∈ Hs × Hs−1 for large enough s, and then to derive Strichartz esti...

Following a discussion of the relation of these problems to applications , intended to clarify the considerations which must be handled in order to obtain genuinely useful results, we consider techniques for determining optimal approximationss and consequent optimal error bounds for certain classes of ill-posed problems with appropriate a priori information.

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H(R) for −3/10 < s.

We study the nonhogeneous heat equation under the form: u t ? u xx = '(t)f(x), where the unknown is the pair of functions (u; f). Under various assumptions about the the function ' and the nal value in t = 1 i.e. g(x), we propose diierent regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For ' 6 6 0 the solution is unique. I. Introducti...

Here u(x, t) represents the free surface of the liquid and the parameter γ > 0 measures the effect of rotation. (1.1) describes the propagation of internal waves of even modes in the ocean; for instance, see the work of Galkin and Stepanyants [1], Leonov [2], and Shrira [3, 4]. The parameter β determines the type of dispersion, more precisely, when β < 0, (1.1) denotes the generalized Ostrovsky...

We consider the motion of a thin filament of viscous fluid in a HeleShaw cell. The appropriate thin film analysis and use of Lagrangian variables leads to the Cauchy-Riemann system in a surprisingly direct way. We illustrate the inherent ill-posedness of these equations in various contexts.

In this paper, we prove the boundedness of Riesz transforms ∂j(−∆) (j = 1, 2, · · · , n) on the Q-type spaces Qα(R n). As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in Q α (R ).

In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves SautTzvetkov’s one and our global well-posedness gives an affirmative answer to SautTzvetkov’s L-data conjecture.

We establish the local well-posedness of the modified Schrödinger map in H3/4+ε(R2).

In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in TriebelLizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the general...