Existence of Solutions to Time-Dependent Nonlinear Diffusion Equations via Convex Optimization

Gabriela Marinoschi ()
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Gabriela Marinoschi: Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy

Journal of Optimization Theory and Applications, 2012, vol. 154, issue 3, No 4, 792-817

Abstract: Abstract This paper aims at providing new existence results for time-dependent nonlinear diffusion equations by following a variational principle. More specifically, the nonlinear equation is reduced to a convex optimization problem via the Lagrange–Fenchel duality relations. We prove that, in the case when the potential related to the diffusivity function is continuous and has a polynomial growth with respect to the solution, the optimization problem is equivalent with the original diffusion equation. In the situation when the potential is singular, the minimization problem has a solution which can be viewed as a generalized solution to the diffusion equation. In this case, it is proved, however, that the null minimizer in the optimization problem in which the state boundedness is considered in addition is the weak solution to the original diffusion problem. This technique allows one to prove the existence in the cases when standard methods do not apply. The physical interpretation of the second case is intimately related to a flow in which two phases separated by a free boundary evolve in time, and has an immediate application to fluid filtration in porous media.

Keywords: Convex optimization problems; Variational inequalities; Brezis–Ekeland principle; Nonlinear diffusion equations; Porous media equation (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-012-0017-6

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