Stability of self-similar extinction solutions for a 3D Darcy flow suction problem

2009 ◽  
Vol 20 (4) ◽  
pp. 343-362 ◽  
Author(s):  
ERWIN VONDENHOFF ◽  
GEORG PROKERT
Keyword(s):  
Surface Tension ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Single Point ◽  
Stability Result ◽  
Darcy Flow ◽  
Moving Boundary Problem ◽  
Axially Symmetric ◽  
Moving Domains ◽  
Self Similar

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties.

scholarly journals A moving boundary problem for the Stokes equations involving osmosis: Variational modelling and short-time well-posedness

2015 ◽  
Vol 27 (4) ◽  
pp. 647-666
Author(s):  
FRIEDRICH LIPPOTH ◽  
MARK A. PELETIER ◽  
GEORG PROKERT
Keyword(s):  
Surface Tension ◽  
Boundary Problem ◽  
Osmotic Pressure ◽  
Moving Boundary ◽  
Stokes Equations ◽  
Semipermeable Membrane ◽  
Moving Boundary Problem ◽  
Classical Solutions ◽  
Well Posedness ◽  
Short Time

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short-time.

Analyticity of solutions to a multidimensional moving boundary problem modelling tumour growth

2011 ◽  
Vol 141 (6) ◽  
pp. 1317-1336 ◽  
Author(s):  
Fujun Zhou ◽  
Junde Wu
Keyword(s):  
Surface Tension ◽  
Internal Pressure ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Moving Boundary Problem ◽  
Regularity Of Solutions ◽  
Order Partial Differential Equation ◽  
Non Local ◽  
Model Equations ◽  
The Given

We consider the regularity of solutions to a multidimensional moving boundary problem modelling the growth of non-necrotic solid tumours. The model equations include two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure within the tumour, respectively, and a first-order partial differential equation governing the evolution of the moving boundary on which surface tension effects counteract the internal pressure. On account of the moving boundary and surface tension effects, this problem is a nonlinear problem involving non-local terms. By employing the functional analytic method and the theory of maximal regularity, we prove that the moving boundary is real analytic in temporal and spatial variables, even if the given initial data admit less regularity.

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