scholarly journals Free boundaries in problems with hysteresis

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140271 ◽  
Author(s):  
D. E. Apushkinskaya ◽  
N. N. Uraltseva
Keyword(s):  
Free Boundary ◽  
Sobolev Class ◽  
Free Boundary Problems ◽  
Strong Solutions ◽  
Chemical Processes ◽  
Free Boundaries ◽  
Open Problems ◽  
Boundary Problems ◽  
Hysteresis Operator ◽  
With Memory

Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes ‘with memory’ in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called ‘strong solutions’ belonging to the anisotropic Sobolev class with sufficiently large q . Several open problems in this direction are proposed as well.

scholarly journals Free boundary problems in shock reflection/diffraction and related transonic flow problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140276 ◽  
Author(s):  
Gui-Qiang Chen ◽  
Mikhail Feldman
Keyword(s):  
Free Boundary ◽  
High Speed ◽  
Free Boundary Problems ◽  
Fluid Flows ◽  
Shock Reflection ◽  
Two Dimensional ◽  
Open Problems ◽  
Boundary Problems ◽  
Flow Problems ◽  
Concave Corner

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

Unsteady two-dimensional flows with free boundaries. - I. General theory

1956 ◽  
Vol 235 (1202) ◽  
pp. 375-381 ◽  
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Steady State Solution ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Boundary Problems ◽  
Initial Development ◽  
Bernoulli’S Equation ◽  
Hodograph Plane ◽  
Two Dimensional Flows

The paper contains a summary of relevant earlier work on free-boundary problems (§1) and then considers the initial development of steady two-dimensional flows. The motion of an incompressible in viscid fluid with free boundaries is considered (§2) by transforming into the hodograph plane (In q 0 / U , θ 0 ) of the steady flow. The equation of the free boundary and the velocity potential are expanded in powers of e- λt . Thus ϕ ( x, y, t ) = ϕ 0 ( x, y ) + e - λt ϕ 1 ( x, y ) + ..., where ϕ 0 is the known steady-state solution and ϕ 1 is to be determined. The exact boundary condition, which is the unsteady form of Bernoulli’s equation, is applied on the free boundary which is not taken as a streamline. A general discussion of the validity of the approach is given (§3). It is foreshadowed that for jet flow through a slit the predicted shape of the jet will probably have a kink at the nose; this is consistent with the assumptions made in the analysis.

scholarly journals Two-Phase Free Boundary Problems: From Existence to Smoothness

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Daniela De Silva ◽  
Fausto Ferrari ◽  
Sandro Salsa
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Free Boundary Problems ◽  
Elliptic Operators ◽  
Free Boundaries ◽  
Two Phase ◽  
Boundary Problems ◽  
Open Questions

AbstractWe describe the theory we developed in recent times concerning two-phase free boundary problems governed by elliptic operators with forcing terms. Our results range from existence of viscosity solutions to smoothness of both solutions and free boundaries. We also discuss some open questions, possible object of future investigation.

ON THE EXISTENCE OF SPATIALLY PATTERNED DORMANT MALIGNANCIES IN A MODEL FOR THE GROWTH OF NON-NECROTIC VASCULAR TUMORS

2001 ◽  
Vol 11 (04) ◽  
pp. 601-625 ◽  
Author(s):  
AVNER FRIEDMAN ◽  
FERNANDO REITICH
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Radial Symmetry ◽  
Vascular Tumors ◽  
Free Boundaries ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Equilibrium Configurations ◽  
Growth Equilibrium

Despite their great importance in determining the dynamic evolution of solutions to mathematical models of tumor growth, equilibrium configurations within such models have remained largely unexplored. This was due, in part, to the complexity of the relevant free boundary problems, which is enhanced when the process deviates from radial symmetry. In this paper, we present the results of our investigation on the existence of non-spherical dormant states for a model of non-necrotic vascularized tumors. For the sake of clarity we perform the analysis on two-dimensional geometries, though our techniques are evidently applicable to the full three-dimensional problem. We rigorously show that there is, indeed, an abundance of steady states that are not radially symmetric. More precisely, we prove that at any radially symmetric stationary state with free boundary r=R0 (which we first show to exist), there begin infinitely many branches of equilibria that bifurcate from and break the symmetry of that radial state. The free boundaries along the bifurcation branches are of the form [Formula: see text], where ℓ=2,3,… and |ε|<ε0; each choice of ℓ and ε determines a non-radial steady configuration.

Numerical treatment of a singularity in a free boundary problem

1972 ◽  
Vol 330 (1583) ◽  
pp. 573-580 ◽  
Keyword(s):  
Free Boundary ◽  
Numerical Solution ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Complex Variable ◽  
Free Boundary Problems ◽  
Separation Point ◽  
Free Boundaries ◽  
Boundary Problems ◽  
A Free Boundary Problem

The numerical solution of free boundary problems gives rise to many computational difficulties. One such difficulty is due to the singularity at the separation point between the fixed and free boundaries. A method is suggested which uses complex variable techniques to determine the shape of the free boundary near to the separation point. This complex variable solution is also used to improve the accuracy of the finite-difference solution in the neighbourhood of the singularity. The analytical study was incorporated into an algorithm for the numerical solution of a particular free boundary problem concerning the percolation of a fluid through a porous dam. Some numerical results for this problem are presented.

scholarly journals An overview of unconstrained free boundary problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140281 ◽  
Author(s):  
Alessio Figalli ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Free Boundary Problems ◽  
Regularity Of Solutions ◽  
Regular Solutions ◽  
Optimal Regularity ◽  
Open Problems ◽  
Boundary Problems ◽  
Matching Problems ◽  
Open Set

In this paper, we present a survey concerning unconstrained free boundary problems of type where B 1 is the unit ball, Ω is an unknown open set, F 1 and F 2 are elliptic operators (admitting regular solutions), and is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

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