The Null Controllability Of Single/Double Coupled Degenerate Parabolic Systems With General Convection Terms

In this thesis,we study the null controllability of the single/double coupled degen-erate parabolic systems with general convection terms.This thesis is divided into two parts.In the first part,we study the null controllability of the single coupled degen-erate parabolic system with general convection terms.In the single coupled parabolic system we studied,the equations degenerate on the boundary and the convection terms are not controlled by the diffusion terms.The convection terms can be nonlinear and the source terms can also be nonlinear in the weakly degenerate case.It is necessary to add restrictions to the coefficients of the convection terms in the relatively strongly degenerate case,and the convection terms are linear,while the source terms can be nonlinear.We divide the first part into two chapters according to the different re-strictions on the coefficients of the convection terms due to the different degeneracy degrees.The second part considers the null controllability of the double coupled linear degenerate parabolic system with general convection terms,and the convection terms are not controlled by the diffusion terms.In the first chapter,we study weakly degenerate case,that is,the null controllabil-ity of the single coupled degenerate parabolic system with nonlinear convection terms and source terms.In the previous work about the null controllability of the single cou-pled degenerate parabolic systems,there is no result that the convection terms are not controlled by the diffusion terms,while we consider the case that the convection terms are not controlled by the diffusion terms.As is well known,the key to the proof of the null controllability is to establish the Carleman estimate of the conjugate problem of the problem,and the convection terms need to be controlled by the diffusion terms when establishing the Carleman estimate.Therefore,when the equations degenerate on the boundary,if their convection terms also degenerate on the boundary,the exis-tence of the convection terms will not cause essential difficulties to the establishment of the Carleman estimate.However,the existence of general convection terms will bring completely essential difficulties to the establishment of the Carleman estimate.Thus,it is particularly important to choose the appropriate energy weight functions to establish the Carleman estimate.Since the coefficients of diffusion terms in single coupled degenerate parabolic equations we considered may be different,it is necessary to choose different energy weight functions for different equations to establish the Car-leman estimate needed by the single coupled system.The Carleman estimate needed by the single coupled degenerate system can not be obtained by directly applying the Carleman estimate and energy weight function of a single equation.In order to over-come this difficulty,we need to make some adjustments to the energy weight functions and obtain the Carleman estimate needed by the single coupled system through some complicated estimates.In this chapter,we firstly linearize and regularize the nonlinear single coupled degenerate system,establish the uniform Carleman estimate of its con-jugate problem,and obtain the observability inequality from the Carleman estimate.Then,we obtain the null controllability of the regularize nonlinear single coupled sys-tem through the observability inequality and the fixed point theorem.Finally,we get the null controllability of the nonlinear single coupled degenerate system through a limit process.The second chapter considers the relatively strongly degenerate case,that is,the null controllability of the single coupled degenerate parabolic system with linear con-vection terms and nonlinear source terms.We still study the case that the convection terms are not controlled by the diffusion terms,but unlike the previous chapter,the degeneracy in this chapter is relatively strong.Since the degeneracy here is relatively strong,if we use the energy weight function and the method in the previous chapter to establish the Carleman estimate,the existence of general convection terms will cause some estimates can not be offset.Therefore,we need to choose new energy weight func-tions to overcome the difficulty brought by the existence of general convection terms to establish the Carleman estimate for the single coupled degenerate system,which requires us to increase the restrictions on the coefficients of convection terms.Since the coefficients of diffusion terms in single coupled degenerate parabolic equations we studied may be different,it is necessary to choose different energy weight functions for different equations when establishing the Carleman estimate needed by the single coupled system.Directly applying the Carleman estimate of a single equation cannot establish the Carleman estimate needed by the single coupled system,so we need to adjust the energy weight function and the Carleman estimate,and then establish the Carleman estimate needed by the single coupled degenerate system through some ac-curate and complicated estimates.In this chapter,we firstly establish the Carleman estimate and the observability inequality of conjugate system of the single coupled de-generate system.Then,the observability inequality can prove the null controllability of the single coupled degenerate system.In the third chapter,we discuss the null controllability of the double coupled linear degenerate parabolic system with general convection terms.As for the results of the null controllability of the double coupled degenerate parabolic systems with the convection terms,the convection terms are controlled by the diffusion terms,and there is no result that the convection terms are not controlled by the diffusion terms,while we consider the case that the convection terms are not controlled by the diffusion terms.Different from the proof of the null controllability of the single coupled system,the null controllability of the double coupled system we considered can not be obtained by directly applying the observability inequality.In order to overcome this difficulty,we need firstly to prove the null controllability of the double coupled system with two control functions,and then the control function of the double coupled system we considered is constructed by the solution and control functions of the double coupled system with two control functions.In this chapter,it is still necessary to choose appropriate energy weight functions to overcome the difficulty of establishing Carleman estimate caused by the existence of general convection terms.For the double coupled system we considered,the coefficients of diffusion terms of different equations may be different,so the Carleman estimate can be established by choosing different energy weight functions for different equations.Firstly,we apply the Carleman estimate of a single equation and establish the Carleman estimate needed by the double coupled degenerate system with two control functions after some accurate estimates.Then,we derive the observability inequality from the Carleman estimate,and then obtain the null controllability of the double coupled system with two control functions.Finally,the control function of the double coupled degenerate system with one control function is constructed by the solution and the control functions of the double coupled degenerate system with two control functions.