On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core

In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826-847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem.     

Finite element analysis of compressible and incompressible fluid-solid systems

This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem. Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

Mejora de la solución fuertemente acoplada de problemas FSI mediante una aproximación de la matriz tangente de presión

The interaction of fluids with surrounding structures constitutes a classical challenge for the different numerical techniques. The aim of current work is twofold: first we provide a simple theoretical explanation of the problems to be faced in incompressible FSI. Then we introduce and justify a new procedure for the solution of complex fluid-structure interaction problems. Such a new strategy is based on the introduction of an «interface Laplacian» at the coupling boundary. The idea is to consider the dependence between fluid pressure and structural velocity as a non linear problem for which a Quasi-Newton scheme is sought. The new interface term is then proved to be an approximation of the tangent matrix for such non-linear problem. In the derivation of this result we make use exclusively of discrete linear algebra. Finally, we prove the efficiency of the new approach showing its ability to tackle standard benchmark problems.     

Interface evolution: Water waves in 2-D

We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The well-posedness of the full water wave problem was first obtained by Wu (1997) [20]. The methods introduced in this paper allow us to consider multiple cases: with or without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It is assumed that the initial interface does not touch itself, being a part of the evolution problem to check that such property prevails for a short time, as well as it does the Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from the mere fact that we are dealing with weak solutions of Euler's equation in the whole space.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Perturbation of semi-linear evolution equations under weak assumptions at initial time

We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi-linear parabolic and hyperbolic boundary value problems with various boundary conditions. The theory also implies the well known theory on parameter dependent equations.     

Necessary Conditions for the Extremum in a Variational Problem on Phase Transitions with Nonhomogeneous Boundary Conditions

A variational problem on phase transitions in elastic media with nonhomogeneous boundary conditions is considered. Necessary conditions for a local minimum of the energy functional are established. These conditions are derived in the weak form of some integral identity, as well as in the form of the classical equilibrium equations. In the first case, no additional smoothness of the solution is required, whereas, in the second case, some additional conditions on the smoothness of the replacement field and the boundary of the interface of the phases are imposed. As was shown, even in the case of nonhomogeneous boundary conditions, the boundary of the interface of the phases intersects the boundary of the domain occupied by an elastic medium only at right angles. Bibliography: 3 titles.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.     

Front-fixing FEMs for the pricing of American options based on a PML technique

In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods.     

A Free Boundary Problem Governed by Nonlinear Degenerate Equations of Parabolic Type

We consider a free boundary problem connected with non-Newtonian fluid motion, i.e. the flow of power law fluids with the yield stress. We obtain the solution of the relevant approximation problem by means of a parabolic quasi-variational inequality, and then obtain the weak solution of the original problem after a passage to the limit. Finally, we study the regularity of the weak solution.     

Global existence and blowup of solutions to a free boundary problem for mutualistic model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained. The asymptotic behavior of the free boundary problem is studied. Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.     

一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.     

Backward Euler method for abstract time-dependent parabolic equations with variable domains

We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial value problem where , , is a family of sectorial operators in a Banach space X. The domains are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the , , norms. We prove that the semidiscretization is stable in a suitable sense. We get optimal estimates for the error even when non-homogeneous boundary values are considered. In particular, the results are applicable to the analysis of the semidiscretizations of time-dependent parabolic problems under non-homogeneous Neumann boundary conditions. Received October 17, 1997 / Revised version received April 17, 1998     

Liquid Flow in Partially Saturated Porous Media

We consider a picture for the filtration of a liquid in a partiallysaturated porous medium, leading to a two-phase one-dimensionalfree boundary problem of the following type: The liquid pressuresatisfies an elliptic equation in the saturated region and anon-linear parabolic equation in the unsaturated region, whilepressure and velocity are continuous across the interface. This scheme reduces to the study of the non-linear parabolicfree boundary problem in the unsaturated phase with cauchy dataprescribed on the free boundary, for such a problem existence,uniqueness and continuous dependence theorems are proved.     

The inverse problem with free boundary for a weakly degenerate parabolic equation

In a domain with free boundary, we consider the inverse problem of determination of a time-dependent coefficient of the first derivative of the unknown function in a parabolic equation with weak power degeneracy. The integral conditions are given as overdetermination conditions. The conditions of existence and uniqueness of the classical solution to this problem are established.     

Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.     

Stefan problem for a weakly degenerate parabolic equation

In a domain with free boundary, we consider the inverse problem of determination of the coefficient of the first derivative of the unknown function in a parabolic equation with weak power degeneration. The Stefan condition and the integral condition are used as overdetermination conditions. The conditions for existence and uniqueness of the classical solution of the posed problem are established.