We study the existence and uniqueness of solutions of the convective-diffusive elliptic equationUnder the assumption that V ∈ L p ( ) N with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u ∈ H 1 ( ) if f dx = 0, and also that the kernel is generated by a function u ∈ H 1 ( ), unique up to a multiplicative constant, which satisfies u > 0 a.e. on . We also prove that the equation −div(D∇u) + div(V u) + ν u = f has a unique solution for all ν > 0 and the map f → u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equationThe dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.
Mathematics Subject Classification (2000)35D05 · 35B30 · 35J25 · 35K20 · 47B44
In this paper we investigate the movement of free boundaries in the two-dimensional Hele-Shaw problem. By means of the construction of special solutions of self-similar type we can describe the evolution of free boundary corners in terms of the angle at the corner. In particular, we prove that, in the injection case, while obtuse-angled corners move and smooth out instantaneously, acute-angled corners persist until a (finite) waiting time at which, at least for the special solutions, they suddenly jump into an obtuse angle, precisely the supplement of the original one. The critical values of the angle π and π/2 are also considered.
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