On uniqueness techniques for degenerate convection-diffusion problems
by Boris Andreianov; Noureddine Igbida
International Journal of Dynamical Systems and Differential Equations (IJDSDE), Vol. 4, No. 1/2, 2012

Abstract: We survey recent developments and give some new results concerning uniqueness of weak and renormalised solutions for degenerate parabolic problems of the form u_t – div(a₀ (∇w) + F(w)) + f, u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a₀, a continuous convection flux F, and an initial condition u|₀ = u₀. The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space. We focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the 'parabolic-elliptic aspects'. We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of 'going to the boundary' in the Kato inequality for comparison of two solutions; uniqueness for renormalised solutions obtained via reduction to weak solutions. On several occasions we use the nonlinear semigroup theory.

Online publication date: Wed, 10-Dec-2014

The full text of this article is only available to individual subscribers or to users at subscribing institutions.

 
Existing subscribers:
Go to Inderscience Online Journals to access the Full Text of this article.

Pay per view:
If you are not a subscriber and you just want to read the full contents of this article, buy online access here.

Complimentary Subscribers, Editors or Members of the Editorial Board of the International Journal of Dynamical Systems and Differential Equations (IJDSDE):
Login with your Inderscience username and password:

Username:        Password:          Forgotten your password?

Want to subscribe?
A subscription gives you complete access to all articles in the current issue, as well as to all articles in the previous three years (where applicable). See our Orders page to subscribe.

If you still need assistance, please email subs@inderscience.com