BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

Estimates for the Dirichlet-wave Equation and Applications to Nonlinear Wave Equations

We shall mainly concern ourselves with the physically important case where the spatial dimension n equals 3. It is considerably easier to prove estimates for the wave equation in odd-spatial dimensions in part because of the fact that the sharp Huygens principle holds in this case for solutions of the boundaryless wave equation in Minkowski space R+×R. By this we mean that if v solves the Minko...

Bilinear Estimates and Applications to 2d Nls

The three bilinearities uv, uv, uv for functions u, v : R2×[0, T ] 7−→ C are sharply estimated in function spaces Xs,b associated to the Schrödinger operator i∂t+∆. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cu...

Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations

We prove a quadrilinear integral estimate in space-time for solutions of the homogeneous wave equation on R. This estimate is a generalization of a previously known bilinear L estimate, and it arises naturally in the study of the local regularity properties of a hyperbolic model equation connected with wave maps from Minkowski space R into a sphere. The scale invariant data space for this equat...

Anisotropic Bilinear L Estimates Related to the 3d Wave Equation

We first review the L2 bilinear generalizations of the L4 estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions influence the estimates. Moreover, we p...

Introduction to Nonlinear Wave Equations