Related papers: Free upper boundary value problems for the semi-ge…
In this paper, we develop a general theory for the transport equation within the framework of Triebel-Lizorkin spaces. We first derive commutator estimates in these spaces, dispensing with the conventional divergence-free condition, via the…
We consider a free boundary problem of compressible-incompressible two-phase flows with phase transitions in general domains of $N$-dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The…
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a…
We present a set of effective outflow/open boundary conditions and an associated algorithm for simulating the dynamics of multiphase flows consisting of $N$ ($N\geqslant 2$) immiscible incompressible fluids in domains involving outflows or…
In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate…
We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a generic phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The…
In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method…
We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was…
In this paper, we study a free boundary problem for a class of nonlinear nonautonomous size structured population model. Using the comparison principle and upper lower solution methods, we establish the existence of the solution for such…
We prove that the 3-D compressible Euler equations with surface tension along the moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and…
The problem of incompressible fluid mixing arises in numerous engineering applications and has been well-studied over the years, yet many open questions remain. This paper aims to address the question "what do efficient flow fields for…
A global existence result is established for a free boundary problem of planar magnetohydrodynamic fluid flows with radiation and large initial data. Particularly, it is novelty to embrace the constant transport coefficient. As a…
We present a formal derivation of the inviscid 3D quasi-geostrophic system (QG) from primitive equations on a bounded, cylindrical domain. A key point in the derivation is the treatment of the lateral boundary and the resulting boundary…
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the…
In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is…
This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical…
In this paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain \Omega. By using the contraction mapping principle and operator semigroup approach, we establish the…
We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of…
Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the…
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost…