Global Well-posedness And Long Time Behavior For Some Nonlinear Dispersive Equations

Evolution equations describe the states or processes in physics and other scientificfields that evolve as the time goes, and are a collective name of many important partial dif-ferential equations that depend on the time variable. Lots of partial di?erential equationsthat model complex phenomena are nonlinear. Nonlinear partial di?erential equationsare the frontier and hot topic of nonlinear sciences. Nonlinear dispersive equations area class of important nonlinear evolution equations. Nonlinear dispersive equations andtheir combinations with the wave equations are used to describe the wave phenomena inphysics, e.g., acoustic waves on a crystal lattice, ion-acoustic waves in a plasma, signalin optical fiber. In the past few decades, the theory of nonlinear dispersive equationsdevelops rapidly. In this dissertation we study the well-posedness and long time behaviorfor some nonlinear dispersive equations.In Chapter 2, we consider the global well-posedness for the following fifth-orderCamassa-Holm equationin Sobolev spaces H~s(R) for s < 1. By I-method we prove that the equation is globallywell-posedness in H~s(R) for s > (6√10 ? 17)/4≈0.493. In order to obtain lower indexfor global well-posedness, on the one hand, we establish bilinear estimates involving theoperator I and computer the exact relationship between the lifespan of the local solutionand the initial data, on the other hand, we prove the almost conservation law by delicateargument. It is worth noting that the global solution obtained by I-method increasespolynomially in time. Compared with the KdV equation, this equation is more complexand lacks scaling invariance, so the range of the index for the global well-posedness weobtained is smaller than that for the local one.In Chapter 3, we consider the global well-posedness for the periodic fifth-orderCamassa-Holm equation in (H|˙)~s(T) for s < 1. By I-method, we prove that the equa-tion is global well-posedness in (H|˙)~s(T) for s > 2/3. In order to control the Lt∞Hxs norm ofthe solution, we introduce the function space Ys and establish the corresponding bilinearestimates, which is di?erent from the Cauchy problem.In many real situations, we cannot neglect energy dissipation mechanisms and ex-ternal excitation, especially for the long time behavior. Therefore, in this dissertationwe also consider the global attractor for the following weakly damped forced modified Camassa-Holm equationwhereλ> 0.In Chapter 4, we consider the global attractor for the weakly damped forced modifiedCamassa-Holm equation in H1(R). Assuming that the forcing term f∈H1(R), we provethat the solution operator to the equation possesses a global attractor in H1(R), andmoreover it is compact in H4(R). In the proof, we split the solutions into two parts,one of which is uniformly bounded in H4(R) and the other decays to zero in H1(R) astime goes to infinity. The compactness of the regular part is obtained by energy equationmethod and extensively exploiting the dispersive regularization properties of the equation.The global solution obtained by I-method increases polynomially in time while theweakly damping termλu may prevent the solution from increasing unlimitedly, thereforewe can expect that the existence of the global attractor for the weakly damped forcedmodified Camassa-Holm equation in (H|˙)~s(T) for s < 1. In Chapter 5, we consider theweakly damped forced modified Camassa-Holm equation in (H|˙)~s(T) for s < 1. Due tothe idea of I-method and under the assumption f∈H˙1(T), we prove that the solutionoperator to the equation possesses a global attractor, and moreover it is compact inH4(T). In the proof, we split the solutions into two parts, one of which is uniformlybounded in H1(T) and the other decays to zero in H~s(R) as time goes to infinity. Thecompactness of the regular part is obtained by the compact embedding from H1(T) intoH~s(T).In Chapter 6, we consider the global attractor for the weakly damped forced KdVequation in (H|˙)~s(T) for s < 0. Under the assumption f∈L˙2(T), we prove that thesolution operator to the equation possesses a global attractor in (H|˙)~s(T) for s≥-1/2,and moreover it is compact in H3(T). The index for the global attractor we obtained isthe same as that for the global wellposedness. This improves the Tsugawa's result.