We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the formwhere u is the vector velocity, P is the pressure, θ is the temperature and μ, p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions.
We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L ∞ bounds for the weak solutions.
The goal of this paper is to study a nonlinear system modeling the heat diffusion produced by Joule effect in an electric conductor. Existence, uniqueness, smoothness, and blowup in particular are studied.
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