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Register a new userLocal well-posedness of the compressible FENE dumbbell model of Warner type
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We consider a dilute suspension of dumbbells joined by a finitely extendible nonlinear elastic (FENE) connector evolving under the classical Warner potential $U(s)=-frac{b}{2} log(1-frac{2s}{b})$, $sin[0,frac{b}{2})$. The solvent under consideration is modelled by the compressible Navier--Stokes system defined on the torus $mathbb{T}^d$ with $d=2,3$ coupled with the Fokker--Planck equation (Kolmogorov forward equation) for the probability density function of the dumbbell configuration. We prove the existence of a unique local-in-time solution to the coupled system where this solution is smooth in the spacetime variables and interpreted weakly in the elongation variable. Our result holds true independently of whether or not the centre-of-mass diffusion term is incorporated in the Fokker--Planck equation.
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We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.
We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the zero flux boundary condition. Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl. Math., 68(5):1304--1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter $b>2$. In this article, we show that for the well-posedness of the microscopic FENE model ($b>2$) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function. Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space. Moreover, such a condition still ensures that the distribution is a probability density. The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.
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.
The FENE dumbbell model consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit $sqrt{b}$, yielding all interesting features near the boundary. In this paper we establish the local well-posedness for the FENE dumbbell model under a class of Dirichlet-type boundary conditions dictated by the parameter $b$. As a result, for each $b>0$ we identify a sharp boundary requirement for the underlying density distribution, while the sharpness follows from the existence result for each specification of the boundary behavior. It is shown that the probability density governed by the Fokker-Planck equation approaches zero near boundary, necessarily faster than the distance function $d$ for $b>2$, faster than $d|ln d|$ for $b=2$, and as fast as $d^{b/2}$ for $0<b<2$. Moreover, the sharp boundary requirement for $bgeq 2$ is also sufficient for the distribution to remain a probability density.
We consider 3D free-boundary compressible elastodynamic system under the Rayleigh-Taylor sign condition. It describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor satisfies the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin [85] by Nash-Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach and also establish the incompressible limit. We apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou-Lindblad elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations in spite of a potential loss of derivative in the source term. We first establish the nonlinear energy estimate without loss of regularity for the free-boundary compressible elastodynamic system. The energy estimate is also uniform in sound speed which yields the incompressible limit. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in boundary energy and analyze higher order wave equations as in the previous works of compressible Euler equations (cf. Lindblad-Luo [60] and Luo [62]) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in [60,62] can still be recovered for a slightly compressible elastic medium by further delicate analysis which is completely different from Euler equations.
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