Formation of singularities in nonlinear dispersive PDEs
Open AccessIn this thesis, we study global behavior of solutions to various nonlinear dispersive partial differential equations, which are important in real-world applications such as nonlinear wave equation (NLW) and nonlinear Klein-Gordon equation (NLKG), relevant to wave propagation and turbulence; nonlinear Schr\"odinger equation (NLS), a model for the basic wave-particle motion and a main ingredient in nonlinear optics, plasma, fluid dynamics, while being responsible for modern internet and communications; nonlinear Hartree equation, modeling Coulomb and other potential fields and relevant for problems in general relativity such as the explosion of Boson stars. In this work we are specifically interested in the behavior of solutions which can only exist for some finite time due to formation of singularities in the solutions. This is one of the crucial topics in the modern nonlinear differential equations as it describes such objects as solitary waves or solitons, instabilities, collapses and blow-ups. We first start with understanding the basic model, the NLS equation in the critical regime, where the scaling is $L^2$-critical and on the same order as the conserved mass. We start with investigation of the stable blow-up dynamics, which has been a key question since 80's for the 2d case. We are able to investigate all dimensions up to 12, first confirming that the $L^2$-critical NLS equations have the generic so-called {\it log-log} blow-up rate from the direct numerical simulations, then providing rigorous analytical justification of the {\it log-log} dynamics in higher dimensions ($d \geq5$), at least for the initial data with the mass slightly larger than the mass of the ground state, for which the spectral conjecture had to be proved. Earlier fundamental works of Galina Perelman and series of works of Merle, Raphael as well as Fibich, Merle, and Raphael gave proof for dimension 1 and numerically-assisted proof in dimensions 2-4. We give a numerically-assisted proof of the spectral property for the dimensions from $d=5$ to $d=12$, and a modification of it in dimensions $d \leq 12$. This, combined with previous results of Merle-Rapha\"el, proves the {\it log-log} stable blow-up dynamics in dimensions $d \leq 10$ and radially stable for $d \leq 12$.We next study the blow-up dynamics of the $L^2$-super-critical NLS equations. First, we investigate the profile equation and extend the result of X.P. Wang of 1990 and Budd et al. from 1999 on the existence of the stationary profile solutions (in the supercritical situation there will be more than one positive profile for given parameters) to higher nonlinearities and higher dimensions, which can serve as the final profiles for the stable blow-up in the super-critical NLS. Using the dynamic rescaling method, we investigate the stable blow-up solutions in the supercritical regime and confirm the square root rate of the blow-up, which occurs without any corrections. Unlike the $L^2$-critical case, this rate is numerically observable, since the blow-up solution converges to one of the profiles given by the profile equation at the focusing level $10^{-5}$. We are able to investigate also energy supercritical regimes, which were not studied previously. Our next task is to study the generalized Hartree (gHartree) equation, which is a modification of the NLS equation except now the nonlinearities are nonlocal and of convolution type, which can completely destroy the previous understanding of singularity formation as the influence on the behavior of the solutions is global. For the $L^2$-critical gHartree equation, we nevertheless observe the {\it log-log} blow-up dynamics, which is based on our numerical simulations and asymptotic analysis. Furthermore, we find the ``adiabatic" regime in the blowup formation, which happens before the {\it log-log} regime. In the cases that we studied, surprisingly, the convolution nonlinearities do not influence that much the dynamics of blow-up, while the analytical tools such as spectral properties and instability directions are being significantly changed, and thus, new analytical tools will need to be developed to further understand the rigorous justification of singularity formation in the critical Hartree equation. Next, we study the blow-up dynamics of the $L^2$-super-critical gHartree equation, which have not been investigated in any mathematical literature before. We first prove the existence of the profile solutions to a stationary Hartree-type profile equation, which then serve as the final blow-up profiles in our numerical simulations. Our numerics show that such profile solutions are not unique (similar to the NLS case), furthermore, we obtain the square root blow-up rate. This also happens at the focusing level $10^{-5}$, and thus, numerically observable.We separately address the question about spectral properties of the linearized operators coming from the nonlinear Schr\"odinger equation, and describe the discrete spectrum of such operators, this might be useful for future analytical studies of the supercritical regime of solutions to the NLS-type equations.Finally, we study the nonlinear Klein-Gordon and nonlinear wave equations. We extend the work of Donninger and Schlag 2011, where the authors studied the 3d cubic NLKG equation and considered various gaussian-type initial data to investigate the long-term behavior of solutions, such as scattering and blow up in future positive time. Previously, such data was used to investigate the long-time behavior of the 3d cubic NLS solutions by Holmer, Platte and Roudenko 2010. In the NLKG equation it was shown that there is a ``tails" of globally existing solutions which eventually scatter to the linear evolution. In our work, we take a step further and not only investigate other dimensions and nonlinearities, but also study the behavior of solutions with numerical scheme of higher accuracy, which, on top of high accuracy, also supports energy conservation. We give reasons for the existence of the scattering ``tails", we also show that while radially symmetric, these equations have blow-up solutions which blow up not necessarily at an origin (if the initial speed is large enough), and show open sets of solutions blowing up with positive or negative amplitudes.
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