We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other app...We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.展开更多
We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solution...We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.展开更多
文摘We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.
基金supported by National Basic Research Program of China(973 Program)(Grant No.2011CB808002)National Natural Science Foundation of China(Grant Nos.11071086,11371152,11401439 and 11128102)+3 种基金the Natural Science Foundation of Guangdong Province(Grant No.S2012010010408)the Foundation for Distinguished Young Talents in Higher Education of Guangdong(Grant No.2014KQNCX162)the University Special Research Foundation for Ph.D Program(Grant No.20104407110002)the Science Foundation for Young Teachers of Wuyi University(Grant No.2014zk06)
文摘We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.
