Classical solutions of the two-phase Stefan-type problem

1987 ◽  
Vol 54 (2) ◽  
pp. 315-324
Author(s):  
Józef Osada
Keyword(s):  
Classical Solutions ◽  
Type Problem ◽  
Two Phase

A two-phase obstacle-type problem for the p-Laplacian

2008 ◽  
Vol 35 (4) ◽  
pp. 421-433 ◽  
Author(s):  
Anders Edquist ◽  
Erik Lindgren
Keyword(s):  
Type Problem ◽  
Two Phase

Semi-classical solutions for Kirchhoff type problem with a critical frequency

2020 ◽  
pp. 1-38 ◽  
Author(s):  
Qilin Xie ◽  
Xu Zhang
Keyword(s):  
Variational Method ◽  
Critical Frequency ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Classical State ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

Abstract In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

Generalizing the modified Buckley–Leverett equation with TCAT capillary pressure

2017 ◽  
Vol 29 (2) ◽  
pp. 338-351 ◽  
Author(s):  
K. SPAYD
Keyword(s):  
Capillary Pressure ◽  
Traveling Wave ◽  
Flow In Porous Media ◽  
Classical Solutions ◽  
Wave Analysis ◽  
Two Phase ◽  
Dynamic Capillary Pressure ◽  
Rate Dependent ◽  
Pure Fluid ◽  
Relevant Situation

The Buckley–Leverett partial differential equation has long been used to model two-phase flow in porous media. In recent years, the PDE has been modified to include a rate-dependent capillary pressure constitutive equation, known as dynamic capillary pressure. Previous traveling wave analysis of the modified Buckley–Leverett equation uncovered non-classical solutions involving undercompressive shocks. More recently, thermodynamically constrained averaging theory (TCAT) has generalized the capillary pressure equation by including additional dependence on fluid properties. In this paper, the model and traveling wave analysis are updated to incorporate TCAT capillary pressure as a generalization of dynamic capillary pressure. Solutions of the corresponding Riemann problem are similar to previous results except in the physically relevant situation in which both phases are pure fluids. The results presented here shed new light on the nature of the interface between one pure fluid displacing another pure fluid, in accordance with TCAT.

scholarly journals A two phase boundary obstacle-type problem for the bi-Laplacian

2022 ◽  
Vol 214 ◽  
pp. 112583
Author(s):  
Donatella Danielli ◽  
Alaa Haj Ali
Keyword(s):  
Phase Boundary ◽  
Type Problem ◽  
Two Phase

scholarly journals ON THE DIRICHLET PROBLEM TO ELLIPTIC EQUATION, THE ORDER OF WHICH DEGENERATES AT THE AXIS OF A CYLINDER

2017 ◽  
Vol 22 (5) ◽  
pp. 717-732 ◽  
Author(s):  
Stasys Rutkauskas
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Classical Solutions ◽  
Type Problem ◽  
Smooth Functions ◽  
3 Dimensional ◽  
Dirichlet Type ◽  
Dirichlet Type Problem

In this article, an elliptic equation, which type degenerates (either weakly or strongly) at the axis of 3-dimensional cylinder, is considered. The statement of a Dirichlet type problem in the class of smooth functions is given and, subject to the type of degeneracy, the classical solutions are composed. The uniqueness of the solutions is proved and the continuity of the solutions on the line of degeneracy is discussed.

Classical solutions of the one-dimensional, two-phase Stefan problem

1975 ◽  
Vol 107 (1) ◽  
pp. 311-341 ◽  
Author(s):  
John R. Cannon ◽  
Daniel B. Henry ◽  
Daniel B. Kotlow
Keyword(s):  
Stefan Problem ◽  
Classical Solutions ◽  
Two Phase ◽  
One Dimensional ◽  
The One

scholarly journals On the Uniqueness of the Classical Solutions of the Radial Viscous Fingering Problems in a Hele-Shaw Cell

2001 ◽  
Vol 14 (4) ◽  
pp. 475-482
Author(s):  
Atusi Tani ◽  
Keyword(s):  
Parabolic Equation ◽  
Viscous Fingering ◽  
Classical Solutions ◽  
Two Phase ◽  
Hele Shaw Cell

In [9, 10] we established the existence of classical solutions to two-phase and one-phase radial viscous fingering problems, respectively, in a Hele-Shaw cell by the parabolic regularization and by vanishing the coefficient of the derivative with respect to time in a parabolic equation. In this paper we show the uniqueness of such solutions to the respective problems

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