Global regularity of nonlinear dispersive equations and Strichartz estimates.
Ovcharov, Evgeni Y.
The main part of the thesis is set to review and extend the theory of the so called Strichartztype estimates. We present a new viewpoint on the subject according to which our primary goal is the study of the (endpoint) inhomogeneous Strichartz estimates. This is based on our result that the class of all homogeneous Strichartz estimates (understood in the wider sense of homogeneous estimates for data which might be outside the energy class) are equivalent to certain types of endpoint inhomogeneous Strichartz estimates. We present our arguments in the abstract setting but make explicit derivations for the most important dispersive equations like the Schr¨odinger , wave, Dirac, Klein-Gordon and their generalizations. Thus some of the explicit estimates appear for the first time although their proofs might be based on ideas that are known in other special contexts. We present also several new advancements on well-known open problems related to the Strichartz estimates. One problem we pay a special attention is the endpoint homogeneous Strichartz estimate for the kinetic transport equation (and its generalization to estimates with vector-valued norms.) For example, this problem was considered by Keel and Tao , but at the time the authors were not able to resolve it. We also fall short of resolving that problem but instead we prove a weaker version of it that can be useful for applications. Moreover, we also make a conjecture and give a counterexample related to that problem which might be useful for its potential resolution. Related to the latter is the fact that we now primarily use complex interpolation in the proof of the homogeneous and the inhomogeneous Strichartz estimates, which produces more natural norms in the vector-valued and the abstract setting compared to the real method of interpolation employed in earlier works. Another important direction of the thesis is to study the range of validity of the Strichartz estimates for the kinetic transport equation which requires a separate and more delicate approach due to its vector-valued dispersive inequality and a special invariance property. We produce an almost optimal range of estimates for that equation. It is an interesting fact that the failure of certain endpoint estimates with L∞ or L1-space norms can be shown on characteristics of Besicovitch sets. With regard to applications of these estimates we demonstrate for the first time in the context of a nonlinear kinetic system (the Othmer-Dunbar-Alt kinetic model of bacterial chemotaxis) that its global well-posedness for small data can be achieved via Strichartz estimates for the kinetic transport equation. Another new development in the thesis is connected to the question of the global regularity of the Dirac-Klein-Gordon system in space dimensions above one for large initial data. That question was instigated in the 1970’s by Chadam and Glassey [12, 13, 22] and although a great number of mathematicians have made contributions in the past 30 years, we, together with the independent recent preprint by Gr¨unrock and Pecher , present the first global result for large data. In particular, we prove that in two space dimensions the system has spherically symmetric solutions for all time if the initial data is spherically symmetric and lies in a certain regularity class. Our result is achieved via new inhomogeneous Strichartz estimates for spherically symmetric functions that we prove in the abstract setting and in particular for the wave equation. We make a number of other lesser improvements and generalizations in relation to the Strichartz estimates that shall be presented in the main body of this text.