Abstract: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.
“…This importance reflects directly into a various range of applications. There are applications concerning elastic materials [25], image restoration [7], thermorheological and electrorheological fluids [2,21] and mathematical biology [10].…”
By applying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:is a bounded domain of smooth boundary ∂Ω and ν is the outward normal vector on ∂Ω. p : Ω → R, a : Ω × R N → R N , f : Ω × R → R and g : ∂Ω × R → R are fulfilling appropriate conditions.
“…This importance reflects directly into a various range of applications. There are applications concerning elastic materials [25], image restoration [7], thermorheological and electrorheological fluids [2,21] and mathematical biology [10].…”
By applying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:is a bounded domain of smooth boundary ∂Ω and ν is the outward normal vector on ∂Ω. p : Ω → R, a : Ω × R N → R N , f : Ω × R → R and g : ∂Ω × R → R are fulfilling appropriate conditions.
“…This importance reflects directly into a various range of applications. There are applications concerning elastic materials [22], image restoration [11], thermorheological and electrorheological fluids [4,19] and mathematical biology [13].…”
Using variational methods, we prove in a different cases the existence and multiplicity of a-harmonic solutions for the following elliptic problem:where Ω ⊂ R N (N ≥ 2) is a bounded domain of smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω. f : ∂Ω × R → R, a : Ω × R N → R N , are fulfilling appropriate conditions.
“…Problems with variable exponent growth conditions also appear in the modelling of stationary thermorheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes of filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the ( )-Laplacian can be found in [10][11][12][13][14] and the references therein.…”
In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving ( 1 ( ), 2 ( ))-Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
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