Abstract
Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.
Highlights
Introduction and the main resultsThe evolutionary equation related to the p-Laplacian ut = div a(x)|∇u|p–2∇u (1.1)arises in the fields of mechanics, physics and biology
If a(x) ≡ 1, there is a tremendous amount of work on the existence, the uniqueness and the regularity of the weak solutions of the equation, one can refer to Refs. [1,2,3,4,5,6,7] and the references therein
The second aim of this paper is to prove the stability theorems based on the partial boundary value condition (1.8)
Summary
We can choose χ[τ,s]gn(φm(u – v)) as the test function, s gn φm(u – v) dx dt ∂t s a(x) |∇u|p–2∇u – |∇v|p–2∇v · ∇(u – v)gn φm(u – v) φm(x) dx dt a(x) |∇u|p(x)–2∇u – |∇v|p(x)–2∇v · (u – v)gn φm(u – v) ∇φn dx dt bi(u, x, t) – bi(v, x, t) (u – v)φmxi gn φm(u – v) dx dt bi(u, x, t) – bi(v, x, t) (u – v)xi φngn φm(u – v) dx dt f (u, x, t) – f (v, x, t) gn φm(u – v) dx dt. Since bi(s, x, t) is a Lipschitz function, using Lebesgue dominated convergence theorem, we have s lim n→∞ τ bi(u, x, t) – bi(v, x, t) (u – v)xi φmgn φm(u – v) dx dt = 0,.
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