Shanks Workshop on Mathematical Aspects of Fluid Dynamics

In this talk, I will describe the simplified version of Ericksen and Leslie that models the hydrodynamic flow of nematic liquid crystals, which is a governing equation for the macroscopic continuum description of evolution of the material under the influence of both fluid velocity field and the macroscopic average of the microscopic orientation of rod-like liquid crystal molecules. I will indicate two ways to construct the finite time singularities for such a flow equation for suitably chosen initial data. This is joint work with Tao Huang, Chun Liu, and Fanghua Lin. Irena Lasiecka The University of Memphis, email: [email protected] Construction of global solutions to a 3-D fluid structure interactions with moving interface Abstract: Equations of fluid structure interactions are described by Navier Stokes equations coupled to a dynamic system of elasticity. The coupling is on a free boundary interface between the two regions. The interface is moving with the velocity of the flow. The resulting model is a quasilinear system with parabolic-hyperbolic coupling acting on a moving boundary. One of the main features and difficulty in handling the problem is a mismatch of regularity between parabolic and hyperbolic dynamics. The existence and uniqueness of smooth local solutions has been established by D. Coutand and S. Shkoller Arch. Rational Mechanics and Analysis in 2005. Other local wellposedness results with a decreased amount of necessary smoothness have been proved in a series of papers by I. Kukavica, A. Tuffaha and M. Ziane. The main contribution of the present paper is global existence of smooth solutions. This is accomplished by exploiting either internal or boundary viscous damping occurring in the model. The mismatch of regularity between hyperbolocity and parabolicity is handled by exploiting recently established sharp regularity of the “Dirichlet-Neuman” map for hyperbolic solvers along with maximal parabolic regularity available for Stokes operators. Maximal parabolic regularity allows to determine the pressure from finite energy solutions of the Stokes problem, while sharp trace regularity allows to propagate the boundary traces through the interface without a loss of regularity (standard functional analysis trace theorems in Lp scales provide a loss of 1/p of derivative when restricting a function to the boundary). These two ingredients prevent a typical “loss” of derivatives and allow for the algebraic and topological closure of a suitably constructed fixed point. This work is joint with M. Ignatova (Stanford University), I. Kukavica (University of Southern California, Los Angeles), and A. Tuffaha (The Petroleum Institute, Abu Dhabi, UAE). 1 Equations of fluid structure interactions are described by Navier Stokes equations coupled to a dynamic system of elasticity. The coupling is on a free boundary interface between the two regions. The interface is moving with the velocity of the flow. The resulting model is a quasilinear system with parabolic-hyperbolic coupling acting on a moving boundary. One of the main features and difficulty in handling the problem is a mismatch of regularity between parabolic and hyperbolic dynamics. The existence and uniqueness of smooth local solutions has been established by D. Coutand and S. Shkoller Arch. Rational Mechanics and Analysis in 2005. Other local wellposedness results with a decreased amount of necessary smoothness have been proved in a series of papers by I. Kukavica, A. Tuffaha and M. Ziane. The main contribution of the present paper is global existence of smooth solutions. This is accomplished by exploiting either internal or boundary viscous damping occurring in the model. The mismatch of regularity between hyperbolocity and parabolicity is handled by exploiting recently established sharp regularity of the “Dirichlet-Neuman” map for hyperbolic solvers along with maximal parabolic regularity available for Stokes operators. Maximal parabolic regularity allows to determine the pressure from finite energy solutions of the Stokes problem, while sharp trace regularity allows to propagate the boundary traces through the interface without a loss of regularity (standard functional analysis trace theorems in Lp scales provide a loss of 1/p of derivative when restricting a function to the boundary). These two ingredients prevent a typical “loss” of derivatives and allow for the algebraic and topological closure of a suitably constructed fixed point. This work is joint with M. Ignatova (Stanford University), I. Kukavica (University of Southern California, Los Angeles), and A. Tuffaha (The Petroleum Institute, Abu Dhabi, UAE). 1