A rain water infiltration model with unilateral boundary condition: qualitative analysis and numerical simulations

We present a rigorous mathematical treatment of a model describing rain water infiltration through the vadose zone in case of runoff of the excess water. The main feature of the mathematical problem emerging from the model lies on the boundary condition on the ground surface which is in the form of a unilateral constraint. Existence and uniqueness of a weak solution is proved under general assumptions. We present also the results of a numerical study comparing the proposed model with other models which approach in a different way the rain water infiltration problem. Copyright © 2006 John Wiley & Sons, Ltd.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

On the finite time blow-up for filtration problems with nonlinear reaction

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow-up of solutions subject to sufficiently large initial data provided that the reaction term “overpowers” the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow-up.     

On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions

We give conditions on the nonlinearities of a reaction-diffusion equation with nonlinear boundary conditions that guarantee that any solution starting at bounded initial data is bounded locally around a certain point of the boundary, uniformly for all positive time. The conditions imposed are of a local nature and need only to hold in a small neighborhood of the point .

     

Fixed-sign solutions for a system of singular focal boundary value problems

We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.     

On the dynamics of Navier-Stokes equations for a shallow water model

In this paper, we study a free boundary problem of one-dimensional compressible Navier-Stokes equations with a density-dependent viscosity, which include, in particular, a shallow water model. Under some suitable assumptions on the initial data, we obtain the global existence, uniqueness and the large time behavior of weak solutions. In particular, it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations.     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.     

Existence Uniqueness and Decay of the Global Weak Solutions for a Class of Parabolic Systems with Nonlinear Boundary Conditions

In this paper, a class of parabolic systems with nonlinear boundary conditions is discussed. By introducing a complete metric space with decay of W^s_p-norm, we obiain the existence uniqueness of global weak solutions, our method is simpler than before. A decay estimate of the global weak solutions is obtained simultaneously.     

A free boundary problem in the theory of electrically actuated microdevices

Electrostatically deflected elastic systems are increasingly used in technical applications. We study a typical situation as a free boundary problem and prove, under quite general hypotheses, the existence of a local branch of solutions. Hadamard’s variational formula is crucial in the proof, which is based on the implicit function theorem in Banach spaces.     

The one phase free boundary problem for the -Laplacian with non-constant Bernoulli boundary condition

Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the -Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure on the ``free' streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function is subject to certain convexity properties. In our earlier results we have considered the case of constant . In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the -capacitary potentials in convex rings, with boundaries.

     

On fourth-order boundary value problem with singular data

We consider a simply supported beam with restoring and external forces given as a sum of a continuous function and a Dirac delta distribution. We present sufficient conditions on these data in order to guarantee a unique positive or negative solution, respectively.     

Non-isothermal lubrication problem with Tresca fluid–solid interface law. Part I

We consider the stationary equations for a non-isothermal Newtonian and incompressible fluids, in a three-dimensional bounded domain. The problem is governed by a coupled system involving a balance of linear momentum and the heat energy with Tresca free boundary friction conditions. Existence, uniqueness and regularity of the weak solution to this coupled problem are proved.