The Study Of Well-posedness For Generalized BBM-Burgers Equation In Multi-dimensions

As we know, the Cauchy problem for the evolution equations with large ini-tial perturbation is one of the most difficult problems in the field of equations. The results about the large initial perturbation are not so many. Owing to lack of systematic research methods, we have to start with a scaled equation and find the practicable methods to solve the problem of large initial perturbation. In this thesis, we deliberate the global existence as well as the large time behavior of the Cauchy problem for two scaled BBM-Burgers equations which have dif-ferent dissipative terms. And the BBM-Burgers equation has its own physical background. It is motivated by physical considerations from fluid dynamics. In addition to shallow water waves, the BBM-Burgers equation also can cover acous-tic waves in anharmonic crystals, the Rossby waves in rotating fluids and so on. Under certain conditions, it also can be a model of one-dimensional transmitted waves. Therefore it is applied in semiconductor devices, optical devices and so on. This thesis is arranged as follows:The first chapter is the introduction of the thesis. We first introduce the phys-ical background and the present situation of the research to the BBM-Burgers equation. Later, we also state the organization and the main results. Finally, we explain the symbols and give the preliminary lemmas.In Chapter2, we focus on the global existence as well as the optimal de-cay estimates of the Cauchy problem for the multi-dimensional BBM-Burgers equation with large initial data in the whole-space. The difficulty between large initial data and small initial data is totally different. The key point to figure out the global existence is the uniform L∞boundedness of the solution. Luckily, by making full use of the energy method combined with Fourier analysis method, we get the uniform L∞boundedness of the solution. By the aid of the uniform L∞boundedness, we improve the regularity of the solution. Then we can derive a global solution. For the decay estimates, the normal frequency decomposition method is not working. Therefore, we need to find a new method time-frequency decomposition method to get the L2optimal decay estimates. With the help of the L2optimal decay estimates, L1boundedness of the solution is obtained by using of Green’s function method. Last, we obtain the Lp optimal decay esti-mates of the solution and it’s derivtives by energy method combined with the time-frequency decomposition method.In Chapter3, we continue to study the BBM-Burgers equation in Chapter2. It turns out that from pointwise estimates not only explicit expressions of the time asymptotic behavior of the solution, but also the propagation of the hyperbolic waves can be obtained. Based on this, we hope to get the pointwise estimates of the Cauchy problem for BBM-Burgers equation with large initial perturbation. But due to technical difficulties, we failed. Therefore we have to consider the small initial perturbation around constant state u*. And we also remove the restrictions on the space dimension and the nonlinear term in Chapter2. By the aid of the popular Fourier analysis method, we yield the pointwise estimates of solutions to the nonlinear equation.In Chapter4, we still look for the method to prove the pointwise estimates of the Cauchy problem for BBM-Burgers equation in Chapter2with large ini-tial perturbation. We find out that for the special nonlinear term the point-wise estimates can be obtained. And this also works for the more generalized BBM-Burgers equation. Therefore, in this Chapter, we investigate another BBM-Burgers equation which has more dissipative terms than that in Chapter2. And we yield the global existence, the optimal decay estimates and the pointwise es-timates of the Cauchy problem for the multi-dimensional BBM-Burgers equation with large initial perturbation in the whole-space. As we know, the pointwise es-timates for large initial perturbation is very hard to deal with. The most difficult part is that we can’t use the a prior assumption to deal with the nonlinear term. Luckily, we find that we can make use of the decay estimates of L∞to replace the a prior assumption. We begin first with the Green’s function for the nonlin-ear equation which has more complicated spectrum structure than that of the Green’s function in Chapter2. Later, we employ the time-frequency decomposi- tion method to derive preliminary decay estimates. Then we obtain the L∞decay estimates. By the aid of the decay estimates and Green’s function method, we finally give the pointwise estimates for BBM-Burgers equation with large initial perturbation.