Some Properties Of The Korteweg-de Vries Equation

The Korteweg-de Vries equation was first derived by Korteweg and de Vries in 1895 as a model for propagation of some surface water waves along a channel. The equation furnishes also a very useful approximation model in nonlinear studies whenever one wishes to include and balance a weak nonlinearity and weak dispersive effects. In particular, the equation is now commonly accepted as a mathematical model for the unidirectional propagation of small ampli-tude long waves in nonlinear dispersive systems. In the past fifty years, the Korteweg-de Vries equation has been intensively studied from various aspect-s of both mathematics and physics. It has become the source of important breakthroughs in mechanics and nonlinear analysis and of many developments in algebra, analysis, geometry and physics.The properties of long nonlinear waves propagating horizontally in densi-ty stratified fluids is a topic which has attracted much recent interest. These waves commonly occur on the thermocline of lakes, fjords or coastal waters, and have also been observed on atmospheric inversions. When the total depth of the fluid is small compared to the wavelength, the governing equation is the Korteweg-de Vries equation. When two long internal gravity waves propa-gate with same phase speeds, this phenomenon can be described by a coupled system of two Korteweg-de Vries equations.In this paper, we investigate some properties of the Korteweg-de Vries equation and a coupled system of two Korteweg-de Vries equations, including the Unique Continuation Property, forced oscillations, null controllability with constraints of the Korteweg-de Vries equation and boundary controllability of the coupled system of two Korteweg-de Vries equations.In the first chapter, we talk about the origin of the Korteweg-de Vries equation and the coupled system of two Korteweg-de Vries equations. We introduce the backgrounds of our problems and present some results which have already been known in the literature. Then we give the main results in this paper.In the second chapter, we study the Unique Continuation Property for the Korteweg de-Vries equation posed on a finite interval, namely, if the solution of the Korteweg de-Vries equation equal to zero on an open nonempty subset of the spacial domain, then the solution equal to zero on the whole spatial domain. Unique Continuation Property is an important issue in the theory of partial differential equations. Its history may date back to the classical results of Holmgren and Carleman at the very beginning of the twentieth century. There are many articles concerned with the Unique Continuation Property for the Korteveg-de Vries equation. Compared with the previous results, we need less conditions on the regularity of the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg de-Vries equation. The Carleman estimate for the Korteveg-de Vries equation was also considered in several articles. However, to the best of our knowledge, most of these results are concerned with boundary observation, there are few results about the Carleman estimate with internal observation. Then combining a smoothing property, we prove the Unique Continuation Property we need.In the third chapter, we investigate a damped Korteweg-de Vries equation with forcing on a periodic domain. We can obtain that if the forcing is periodic with small amplitude, then the solution becomes eventually time-periodic. In recent years, the asymptotically time-periodic solutions of the Korteweg-de Vries type equation attracted the attention of many authors. To obtain this result, we introduce Bourgain spaces to overcome this difficulty. According to the bilinear estimate in Bourgain spaces, we can deal with the nonlinear term. Then, applying the properties of Bourgain spaces and a fixed point argument, we give several well-posedness results of the linearized system. Finally, the desired result follows from this well-posedness results and semigroup theory.In the forth chapter, we consider a coupled system of two Korteweg-de Vries equations on a bounded domain. This system is an important model to describe the strong interaction of two long internal gravity waves in a stratified fluid. In previous studies, most of the results focus on the well-posedness and stabilization of this system, in this chapter, we consider this system from the point of control. For the boundary controllability of the Korteweg-de Vries equation, we know that if only the left control input is in action, the equation behaves like a parabolic system and is only null controllable. However, if the equation is allowed to control from the right end of the spacial domain, then the equation behaves like a hyperbolic system and is exactly controllable. Since the exact controllability of the coupled system from the right Dirichlet boundary conditions has been proved, it is natural and interesting to establish the null controllability of this system from the left Dirichlet boundary conditions. First, we transform the system with both linear and nonlinear coupling terms to a system with only nonlinear coupling terms by a change of variable. Next, we prove the null controllability of the linearized system with two distributed controls through a Carleman estimate, then combining the analysis of the linearized system and Kakutani’s fixed point argument, we obtain the null controllability of the original system.In the fifth chapter, we focus on the internal null controllability for the linear Korteweg-de Vries equation with a finite number of constraints on the state. One may come across with this kind of controllability problem while using Lions’s sentinels method to identifying parameters in incomplete data problems. The null controllability problems with constraints on the state and on the control for the heat equation have been widely studied, however, there is no similar result for the third-order dispersion equation. It should be noted that the control problem solved in this chapter is not a straightforward exten-sion to the third-order dispersion equation. Since the Carleman estimate for the third-order linear dispersion equation is not as good as that for the heat equation, it will lead to several difficulties in our proof. In this chapter, we first prove a Carleman inequality adapted to a constraint on the control, then we transform the controllability problem with constraints on the state into an equivalent controllability problem with constraint on the control. Next we construct a symmetric bilinear form and prove that the bilinear form is a scalar product on some space by the adapted Carleman inequality. Finally, we find the control of the equivalent problem according to Lax-Milgram theorem.