Solvability Of Free Boundary Problem Research Articles

Solvability of free boundary problems for steady groundwater flow

In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data.

$$L_p$$ L p -theory of free boundary problems of magnetohydrodynamics in multi-connected domains

We present the \(L_p\)-theory of solvability of free boundary problems of magnetohydrodynamics of viscous incompressible fluids in multi-connected domains constructed in the paper Solonnikov (Interf Free Bound 14:569–603, 2012) for \(p=2\). The case of simply connected domains is studied in Padula and Solonnikov (J Math Sci 178:313–344, 2011), Solonnikov (\(L_p\)-theory of free boundary problems of magnetohydrodynamics in simply connected domains, submitted to AMS Translations).

On the local solvability of free boundary problem for the Navier–Stokes equations

We consider the problem of evolution of a finite isolated mass of a viscous incompressible liquid with a free surface. We assume that the initial configuration of the liquid hasn arbitrary shape, the initial free boundary possesses a certain regularity and the initial velocity satisfies only natural compatibility and regularity conditions (but its smallness is not assumed). We prove that this problem is well posed, i.e., we construct a local in time solution belonging to some Sobolev–Slobodetskii spaces. We expect that this result can be helpful for the analysis of more complicated problems, for instance, problems of magnetohydrodynamics. Bibliography: 9 titles.

Newton’s polygon method and the local solvability of free boundary problems

For some qualitatively new problems with free (unknown) boundary, conditions of local solvability (with respect to time) are investigated.