Smoothing properties of certain dispersive nonlinear partial differential equations

Dispersive partial differential equations;Well-posedness;Smoothing;Zakharov;Klein-Gordon Schrödinger;Majda-Biello;Boussinesq equation

Compaan, Erin Leigh

关键词: Dispersive partial differential equations;    Well-posedness;    Smoothing;    Zakharov;    Klein-Gordon Schrödinger;    Majda-Biello;    Boussinesq equation;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/97315/COMPAAN-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis is primarily concerned with the smoothing propertiesofdispersive equationsand systems. Smoothing in this context means that the nonlinear part of the solution flow is of higher regularity than the initial data. We establish this property, and some of its consequences, for several equations.The first part of the thesis studies a periodic coupled Korteweg-de Vries (KdV) system. This system, known as the Majda-Biello system, displays an interesting dependancy on the coupling coefficient α linking the two KdV equations. Our main result is that the nonlinear part of the evolution resides in a smoother space for almost every choice of α. The smoothing index depends on number-theoretic properties of α, which control the behavior of the resonant sets. We then consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also use a modified energy functional to show that when the damping is large, the attractor is trivial.The next chapter studies the Zakharov and related Klein-Gordon-Schrödinger (KGS) systems on Euclidean spaces. Again, the mainresult is that the nonlinear part of the solution is smoother than the initial data.The proof relies on a new bilinear Bourgain-space estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. As an application, we give a simplified proof of the existence of global attractors for the KGS flow in the energy space for dimensions two and three. We also use smoothing in conjunction with a high-low decomposition to show global well-posedness of the KGS evolution on R4 below the energy space for sufficiently small initial data.In the final portion of the thesis we consider well-posedness andregularity properties of the “good” Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. We also establish a smoothing result, obtaining up to half derivative smoothing of the nonlinear term. Theresults are sharp within the frameworkof the restricted norm method that we use and match known results on the full line.