The Well-posed Problem Of The BBM Equations In Unbounded Domain

In this master’s dissertation, we mainly consider the well-posed problem of BBM (Benjamin-Bona-Mahoney) equations in unbounded domain. With the consideration of many factors, in this paper, I decide to use the locally uniform space to be the phase space.The BBM equation was first proposed as a model for propagation of long waves which incorporates nonlinear dispersive and dissipative effects. And BBM equation are more suitable for mathematical model than the KDV equation.When the spaces region is the R3or a general unbounded domain, The unbounded domains lead the long time behavior of the solution which is the solution of infinite dimensional dissipative dynamical system to very complex. Choosing a suitable phase space is a non-trivial problem when researching on the long-time behavior of the solution. In order to accommodate a constant solution, traveling wave solution and some special forms of solution, locally uniform space is better than the usual Sobolev space and the weighted Sobolev space.We’ll use the method of Galerkin approximation and energy estimation to prove the existence and uniqueness of weak solutions. What is the most important thing of the proof is the estimation of nonlinear term. Since we should use Galerkin approximation to create a point range which is finally approximate to a function, the weak solution of BBM equations, we should make sure that the point range is convergent. Using the estimation of energy can prove that the energy is bounded. The key of the two methods is the growth of the nonlinear terms, and the nonlinear term growing much quickly will likely leads to a divergent energy blasting. Thus the main consideration of the assertation is making approximating sequence convergent, and bounding the energy with nonlinear terms under right conditions. As a result, proving the global existence of weak solution. When getting the global existence, we can use the energy inequality formula to prove the continuous dependence on the initial value.