Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems

We show global well-posedness and exponential stability of equilibria for a general class of nonlinear dissipative bulk-interface systems. They correspond to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes nonlinear slow and fast diffusion in the bulk and nonlinear coupled diffusion on the interface. Additional driving mechanisms can be included and non-smooth geometries and coefficients are admissible, to some extent. An important application are volume-surface reaction-diffusion systems with nonlinear coupled diffusion.Comment: 21 page

Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.Comment: 33 page

Vectorial quasilinear diffusion equation with dynamic boundary condition

summary:In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)$_\varepsilon$ with a nonnegative constant $\varepsilon$, and for any $\varepsilon\ge0$, (S)$_\varepsilon$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma:=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method, which can enable us to handle the transmission systems of various vectorial mathematical models, such as the Bingham type flow equations, the Ginzburg– Landau type equations, and so on. On this basis, we set the goal of this paper to prove two Main Theorems, concerned with the well-posedness of (S)$_\varepsilon$(S) with the precise representation of solution, and $\varepsilon$-dependence of (S)$_\varepsilon$, for $\varepsilon \ge0$

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.Comment: 39 pages, 4 postscript figure

A linear domain decomposition method for partially saturated flow in porous media

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.Comment: 34 pages, 13 figures, 7 table