Abstract
A kind of electrorheological fluid equations with orientated convection terms is considered. If the diffusion coefficient a(x,t)in C^{1}(overline{Q_{T}}) is degenerate on the boundary ∂Ω, not only the uniqueness of weak solution is proved, but also the stability of the solutions can be proved without any boundary condition, provided that there are some restrictions on the diffusion coefficient a(x,t) and the convective coefficient vec{b}(x,t). Moreover, the large time behavior of weak solution is studied.
Highlights
The initial boundary value problem of an electrorheological fluids equation with orientated convection term ut = div a(x, t)|∇u|p(x,t)–2∇u + f (x, t) · ∇uq, (x, t) ∈ QT = Ω × (0, T), (1.1)u(x, 0) = u0(x), x ∈ Ω, (1.2)u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T), (1.3)is studied in this paper, where 1 < p(x, t) ∈ C(QT ), q > 0, a(x, t) ∈ C1(QT ), f = {f i(x, t)}, f i(x, t) ∈ C1(QT ), and Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω.When p(x, t) > 1 is a measurable function on QT, equation (1.1) arises in electrorheological fluids theory [1]
We know that the uniqueness and the stability of weak solutions can be proved if the Dirichlet boundary value condition (1.3) is imposed
We have found that condition (1.5) may replace the usual Dirichlet boundary value condition (1.3) for some special f (x, t, u, ∇u), the stability of solutions can be established without any boundary value condition (1.3), provided that there are some other restrictions on f (x, t, u, ∇u)
Summary
If p(x, t) = p > 1 is a constant, a(x, t) = 1 and f (x, t) = 0, equation (1.1) is well known as non-Newtonian fluid equation and has been studied by many mathematicians, one can refer to [7,8,9,10,11,12,13,14,15] and the references therein From these papers, we know that the uniqueness and the stability of weak solutions can be proved if the Dirichlet boundary value condition (1.3) is imposed. Theorem 1.3 Let q ≥ 1, a(x, t) ≥ 0 satisfy (1.6), p(x, t) ≥ p– > 1, u(x, t) and v(x, t) be two nonnegative weak solutions of equation (1.1) with the initial values u0(x) and v0(x). Theorem 1.4 If q ≥ 1, a(x, t) ≥ 0 satisfies (1.6), p(x, t) ≥ p– > 1, u(x, t) and v(x, t) are two nonnegative weak solutions of equation (1.1) with the initial values u0(x) = v0(x), a(x, t)–.
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