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Register a new userWell-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids
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We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument.
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In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.
A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.
We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a new definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous terms lying in some arbitrary negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.
Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short time.
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