2015-03-25

On the asymptotic analysis of GFDs and the related equationsIn this lecture, we shall introduce time periodic solutions of differential equations arised from different occasions. We then turn our attention to the GFDs and study the asymptotic stabilities of the solutions. In particular, two different approaches will be introduced to prove the existence of time periodic solutions of GFDs with time periodic forces. A few physical implications will be explained. We will also demonstrate some numerical experiments.

A multiscale method to couple network models and continuous equations for two-phase flows in porous media

In this talk, we present a numerical multiscale method for coupling a conservation law for mass at the continuum scale with a discrete network model that describes the pore scale flow in a porous medium. We introduce the dynamical network model that describes two-phase flows in pore scale. We developed single-phase flow algorithms and extended the methods to two-phase flow. Our coupling method for the pressure equation uses local simulations on small sampled network domains at the pore scale to evaluate the continuum equation and thus solve for the pressure in the domain. For local simulation, it often requires a suitable initialization. We introduce a choice of initialization from a optimization problem, which is often used in image processing. We present numerical results for single-phase flows with nonlinear flux- pressure dependence, as well as two-phase flow.

Boundary layer solution to symmetric hyperbolic-parabolic system

We consider the large-time behavior of solutions to the symmetric hyperbolic-parabolic system in the half line. We firstly prove the existence of the stationary solution by assuming that a boundary strength is sufficiently small. We note that the existence of the stationary solution is characterized by a number of negative characteristics. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on the initial perturbation. The key to proof is to derive the uniform a priori estimates by using the energy method in half space developed by Matsumura and Nishida as well as the stability condition of Shizuta–Kawashima type.

Dissipative structure of regularity-loss type for some partial differential equations

In this talk, we discuss dissipative structures for partial differential equations in whole space. For symmetric hyperbolic systems and symmetric hyperbolic-parabolic systems, we had already known the useful characterization of the dissipative structure. Recently, we found some concrete systems which can not be applied this useful result any longer. Furthermore, these concrete systems have a new dissipative structure called regularity-loss structure. Under this situation, our purpose of this talk is to introduce some concrete examples of regularity-loss type and derive the decay property of solutions.

Mathematical analysis of the generalized Bohm criterion

In this talk, we study a boundary layer, called a sheath, which occurs on the surface of materials with which a multicomponent plasma contacts. For the sheath formation, the generalized Bohm criterion demands that the ions enter the sheath region with a high velocity. The motion of the multicomponent plasma is governed by the Euler-Poisson equations, and the sheath is mathematically understood as the stationary solution to the equations. We show the unique existence and the asymptotic stability of the stationary solution under the the generalized Bohm criterion.

2015-03-26

Singularity of macroscopic variables near boundary for gases with cut-off hard potentialIn this talk, the boundary singularity of stationary solutions to the linearized Boltzmann equation with cut-off hard potential is analyzed. A technique of using the H ̈older type con- tinuity of the integral operator to obtain integrability of the derivatives of the macroscopic variables is developed. We establish the asymptotic approximation for the gradient of the moments. Our analysis indicates the logarithmic singularity of the gradient of the moments. In particular, our theorem holds for the condensation and evaporation problem.

Modified scattering of the Vlasov-Poisson system

We study the asymptotic behavior of dispersing solutions to the Vlasov-Poisson system. Due to a long range interaction, we do not expect linear scattering. Instead, we prove a modified scattering result of small and dispersing global solutions. We provide a quasi free forward trajectory and construct its corresponding profile for a given dispersive solution to the Vlasov-Poisson system. The quasi free trajectory consist of two parts. One is linear motion of each particle, and other one is a solution of an ordinary differential equation which represent error correction of long range interaction. The time growth rate of the error correction term is log(t+1) at most.

On a one-dimensional motion of a viscous heat-conducting and self-gravitating gas

We consider the system of equations describing a one-dimensional motion of a viscous, heat- conducting and self-gravitating gas bounded by the free-surface. We first show the temporally global solvability of the problem without any restriction on the size of the initial data. Secondly we obtain the large-time behavior of the flow by establishing uniform in time a priori estimates of the solution under a certain restricted, but physically plausible situation. The asymptotic profile of the flow is given by a particular solution of the corresponding stationary problem.

Motion of a Vortex Filament in an External Flow

In this talk, we consider the motion of a vortex ring under the influence of external flow. This can be seen as an idealization of the motion of a bubble ring traveling through water, where environmental flow is also present.
The motion is described as an initial value problem posed on the one-dimensional torus for a closed vortex filament. The equation of motion is a nonlinear dispersive type equation and is an extension of the Localized Induction Equation (LIE). The LIE is one of the most oldest and fundamental model equation describing the motion of a vortex filament, and the equation we consider in this talk is a generalization of the LIE which takes into account the presence of external flow. The time-local solvability of the initial value problem will be presented, focusing on the derivation of energy estimates needed in order to prove the solvability.

Hydrodynamic limit for the Vlasov-Navier-Stokes equations

The interactions between particles and fluid have received a bulk of attention due to a number of their applications in the field of, for example, biotechnology, medicine, and in the study of sedimentation phenomenon, compressibility of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this talk, we present the rigorous hydrodynamic limit of the Vlasov-Navier-Stokes equations. The limit problem consists of the isothermal Euler equations and the incompressible Navier-Stokes equations. We also discuss the large-time behavior of solutions for the limit problem which shows the exponential alignment between two fluid velocities.

2015-03-27

Global existence for some transport equations with nonlocal velocityIn this talk, we consider nonlocal and quadratically nonlinear transport equations. Proto- typical examples are the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magnetogeostrophic equation and their variants. Among them, we address the global existence of weak solutions of an 1D model of the quasi-geostrophic equa- tion and the full 2D dissipative quasi-geostrophic equations. To this end, we carefully choose dissipative quantities to minimize conditions of initial data using entropies.

Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains

We focus on constructing a vector field whose vorticity, divergence and the normal or the tangential trace are given. In particular, we consider the case that the domain is of Sobolev class, and derive Hodge decomposition type elliptic estimates on domain of such class. Such kind of estimates are useful in the study of free boundary problem involving invicid fluid.

The De-Giorgi’s method as applied to Hamilton-Jacobi type equations and parabolic equations with nonlocal integral operators

In recent years, the study of regularity properties of solutions to time-dependent Hamilton- Jacobi type equations has attracted considerable attention in the community of P.D.E. spe- cialists. In particular, the Holder regularity of viscosity solutions to Hamilton-Jacboi type equations with Hamiltonian satisfying general coercivity properties was first established by P. Cardaliaguet around 2009 through the use of stochastic method. In 2012, the above men- tioned result of P. Cardaliaguet was reproved by P. Cardaliaguet and L. Silvestre through the use of simple comparison principle which is based on the constructions of sub-solutions and super-solutions. In this talk, we will introduce a recent piece of work due to C.H.Chan and A. Vasseur, in which we give an alternative proof of the above mentioned Holder regularity result for solutions to Hamilton-Jacobi type equations. In contrast with the above mentioned work due to P. Cardaliaguet and L. Silvestre, our new proof is based on the De-Girogi’s technique, and uses the coercivity property of the Hamiltonian to induce a parabolic-like regularization effect. In this talk, we will also try to compare the technique used in this alternative proof with another previous work entitled ”Regularity theory for parabolic nonlinear integral operators.” due to L. Caffarelli, C.H.Chan, and A. Vasseur

3-dimensional incompressible Navier-Stokes equations interacting with a nonlinear elastic shell

We study the the Navier-Stokes equations interacting with a nonlinear elastic biofluid shell which can be used to describe the dynamics of the membrane of blood cells. We assume that the interior is occupied by Newtonian fluids while the elastic boundary stores the Willmore energy. The coupling of the two dynamics is through the boundary condition of the PDEs which balances the normal traction forcing due to the fluid and the shell traction due to the deformation of the shell.
To solve the PDEs, we introduce the sign-distance function which measures the distance between the moving boundary and the initial boundary. Using this sign-distance function, we can transform the equations given on the moving domain to equations on a fixed reference do- main, and apply fixed point arguments to construct a solution to the corresponding equations. To explain how we close estimates, I will take a linear case to demonstrate the difficulties and technicality that requires to solve the nonlinear problem.

Space-time Conserved Element and Solution Element Method for Scalar Conservation Laws with Discontinuous Flux in the Space Variable

In this talk, we will study numerical approximation of some scalar conservation laws with discontinuous flux function in the space variable. Indeed, a new modified space time conserved element and solution element method (CESE) is introduced to solve this problem with the given geometric structure of the flux at the interface. The main motivation is to solve this problem by using this modified CESE method which inherits the features of the original CESE method for scalar conservation laws with continuous flux.