Boundary Layer Problem And Vanishing Viscosity Limit Of The Hunter-saxton Equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of a nematic liquid crystal.In a different physical context,it is also the high-frequency limit of the Camassa-Holm equation which arises in the theory of shallow wa-ter waves.It is well-known that the inconsistency of boundary conditions between the viscous parabolic equations and the related inviscid hyperbolic equations usually leads to the boundary layer phenomena.The emergence of the boundary layer is usually accompanied by the problem of the vanishing viscosity limit.In this thesis,we mainly study the boundary layer problem and vanishing viscosity limit of the initial boundary value problem of the Hunter-Saxton equation.The first chapter is the introduction.We briefly introduce the physical background and research progress of Hunter-Saxton equation,and review some existing results on the classical boundary layer theory.Besides,we present the main research contents and some preliminary knowledge of this thesis.In chapter 2,we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation on the half space when the viscosity tends to zero,where the homogeneous Neumann boundary condition is imposed on v~εat the boundary x=0,and show that no boundary layer exists,i.e.,the viscous solutions converge to the inviscid solutions uniformly inε.By using the multi-scale asymptotic expansion,we first formally derive the equations for boundary layer profiles.Next,we establish the well-posedness of the equations for the boundary layer profiles by using the compactness argument.Finally,we construct a suitable approximate solution and obtain the convergence results of the vanishing viscosity limit by the energy method.In chapter 3,we study the vanishing viscosity limit and global-in-time stability of bound-ary layers of an initial boundary value problem for the Hunter-Saxton equation in a bounded domain[0,l],where the Dirichlet boundary condition is imposed on v~εat the boundary x=l.We show that,unlike the findings in chapter 2,there exists a noncharacteristic boundary layer profile near the boundary x=l.We first construct an approximate solution to the viscous Hunter-Saxton equation by the classical multi-scale asymptotic expansion.Then we establish a series of sharp decay estimates for the inner functions and boundary layer functions.Based on these crucial decay estimates and a novel truncated energy method,we finally justify the validity of the expansion and particularly obtain the global-in-time convergence rates as the viscosity shrinks.In chapter 4,we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition.Different from the results of the previous two chapters,we prove that there is a characteristic boundary layer near the boundary x=0.By the formal multiscale analysis,we first derive the characteristic bound-ary layer profile,which satisfies a nonlinear parabolic equation.On the basis of the Galerkin method along with a compactness argument,we then establish the global well-posedness of the boundary layer equation.Finally,we prove the global stability of the boundary layer profiles to-gether with the optimal convergence rates of the vanishing viscosity limit by the energy method.