A variational inequality approach to generalized two-phase Stefan problem in several space variables

Summary The paper offers a study of a broad class of multidimensional two- phase problems of Stefan type by means of variational inequality techniques. The problems for quasilinear equations of alternatively parabolic or mixed parabolicelliptic type, mixed type nonlinear conditions at the fixed lateral boundary, involving free boundary conditions corresponding to phase transitions of both first (latent heat positive) and second kind (latent heat equal to zero) are taken into consideration. Results concerning existence of weak solutions, their uniqueness and stability are established.The preparation of the paper was partially carried out while the author's visiting Istituto di Analisi Numerica del C.N.R., Pavia, due to support of the C.N.R.     

A Stefan problem in combustion

This paper deals with a mathematical model of a condensed two-phase combustion process which describes combustion of solid materials in which melting occurs. The paper shows the existence of a weak solution of the resulting differential equations system and, furthermore, shows that the phase change set (the set where the temperature is equal to the given constant melting temperature) is not a front but a whole mushy region. For this mushy region an estimate in measure is given.This work has been supported by the Deutsche Forschungsgemeinschaft.     

A singular-perturbed two-phase Stefan problem

The asymptotic behavior of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit, the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.     

Stefan problem

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H ^2+α,1+α/2.     

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.     

Existence and uniqueness of weak solutions for precipitation fronts: A novel hyperbolic free boundary problem in several space variables

The determination of the large‐scale boundaries between moist and dry regions is an important problem in contemporary meteorology. These phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families (drying, slow moistening, and fast moistening) of precipitation fronts. The last two wave types violate Lax's shock inequalities yet are robustly realized. This formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit. A new L²‐contraction estimate is also established at positive relaxation values. © 2010 Wiley Periodicals, Inc.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Dirichlet-like space and capacity in complex analysis in several variables

For a Kähler manifold X, we study a space of test functions W^∗ which is a complex version of W1,2. We prove for W^∗ the classical results of the theory of Dirichlet spaces: the functions in W^∗ are defined up to a pluripolar set and the functional capacity associated to W^∗ tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W^∗ is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.     

Solutions for the generalized Loewner differential equation in several complex variables

We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form ${f(z, t)=e^{tA}z+\cdots,}We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z, t)=e^tAz+?,{f(z, t)=e^{tA}z+\cdots,} where A ? L(\mathbbCⁿ, \mathbbCⁿ){A\in L(\mathbb{C}^n, \mathbb{C}^n)} has the property k ₊(A) < 2m(A). Here k₊(A)=max{?l:l ? s(A)}{k_+(A)=\max\{\Re\lambda:\lambda\in \sigma(A)\}} and m(A)=min{?áA(z), z ?: ||z||=1}{m(A)=\min\{\Re\langle A(z), z \rangle: \|z\|=1\}} . (The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results are applied to subordination chains generated by spirallike mappings on the unit ball in \mathbbCⁿ{\mathbb{C}^n} . Finally, we determine the form of the solutions in the presence of certain coefficient bounds.