The Explicit Solution of a Free Boundary Problem Involving Surface Tension

1955 ◽  
Vol 4 (4) ◽  
pp. 557-567 ◽  
Author(s):  
E. McLeod, Jr.
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Explicit Solution ◽  
A Free Boundary Problem

Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

2015 ◽  
Vol 26 (4) ◽  
pp. 401-425 ◽  
Author(s):  
FUJUN ZHOU ◽  
JUNDE WU
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Stationary Solution ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Surface Tension Coefficient ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
A Free Boundary Problem

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Remarks concerning a free boundary problem arising in the theory of liquid drops and in plasma physics

1989 ◽  
Vol 111 (1-2) ◽  
pp. 169-181 ◽  
Author(s):  
John W. Barrett ◽  
Charles M. Elliott
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Plasma Physics ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Liquid Drop ◽  
Liquid Drops ◽  
Plasma Problem ◽  
A Free Boundary Problem ◽  
Surface Tension Coefficients

SynopsisWe consider a generalisation of the liquid drop problem, introduced in [1, Part II], by allowing the upper and lower surfaces to have different surface tension coefficients γv and γu. We study the existence, uniqueness and regularity of this problem. In addition, we show that as γv/γu →0, the solution of this problem converges to the solution of the “plasma problem”.

FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE IDEAL FLUID WITH SURFACE TENSION

2002 ◽  
Vol 12 (12) ◽  
pp. 1725-1740 ◽  
Author(s):  
MASAO OGAWA ◽  
ATUSI TANI
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Incompressible Euler Equation ◽  
Incompressible Ideal Fluid ◽  
Finite Smoothness ◽  
A Free Boundary Problem ◽  
Class Of Functions ◽  
Incompressible Euler

We prove that a free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness. Moreover, it is shown that the solution of this problem converges to the solution of the problem without surface tension as the coefficient of the surface tension tends to zero.

Uniqueness of solution to a free boundary problem from combustion with transport

MAT Serie A ◽  
2001 ◽  
Vol 5 ◽  
pp. 37-41
Author(s):  
Claudia Lederman ◽  
Juan Luis Vázquez ◽  
Noemí Wolanski
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Uniqueness Of Solution ◽  
A Free Boundary Problem
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