Strichartz Estimates And Well-Posed Problems For Nonlinear Wave Equation

This thesis is devoted to the study of the Strichartz estimate and it's angular improve-ment for the linear homogeneous wave equation, the well-posed problems in the Sobolevspace H~s with almost optimal s for derivative semilinear wave equation, and the radialimprovement of the local well-posedness for second order quasilinear wave equation in1+2 dimensions.Firstly, we give an overview and some remarks for the Strichartz estimate for linearhomogeneous wave equation, especially for the L~qL~∞estimate. In addition, under theassumption of spherical symmetry or some further angular regularity assumption for theinitial data, we get the improvement of the Strichartz estimate. These results solves somelong-standing open problems, in particular, we prove a conjecture of Klainerman ([24]).Secondly, based on the Strichartz estimate, we give the local well-posed result inH~s with rough s, for derivative semilinear wave equation. Moreover, for equation withpolynomial nonlinearity in (?)u, we also get the global well-posed result with small data. Incontrast, we study the ill-posedness for the similar equations, illustrating that the positiveresults are almost optimal.Since we get the radial improvement for the Strichartz estimate, we can get the im-provement for the well-posed results for radial initial data. Inspired by this result, we geta local existence result in H~s with rougher s for second order quasilinear wave equationin two space dimensions, by proving a similar Strichartz estimate.