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Register a new userWell-posedness for the stochastic electrokinetic flow
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We consider the stochastic electrokinetic flow in a smooth bounded domain $mathcal{D}$, modelled by a Nernst-Planck-Navier-Stokes system with a blocking boundary conditions for ionic species concentrations, perturbed by multiplicative noise. Several results are established in this paper. In both $2d$ and $3d$ cases, we establish the global existence of weak martingale solution which is weak in both PDEs and probability sense, and also the existence and uniqueness of the maximal strong pathwise solution which is strong in PDEs and probability sense. Particularly, we show that the maximal pathwise solution is global one in $2d$ case without the restriction of smallness of initial data.
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The Cauchy problem of a multi-dimensional ($dgeqslant 2$) compressible viscous liquid-gas two-phase flow model is concerned in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
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.
We present a new proof of well-posedness of stochastic evolution equations in variational form, relying solely on a (nonlinear) infinite-dimensional approximation procedure rather than on classical finite-dimensional projection arguments of Galerkin type.
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L^infty_t L^2_x$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.
The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrodinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrodinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $dgeq 4$.
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