ABSTRACT We state a partial regularity result for steady flows of a class of incompressible fluid... more ABSTRACT We state a partial regularity result for steady flows of a class of incompressible fluids, their viscosity varies with the shear-rate and the pressure. Apart from partial regularity in three dimension we obtain, as a consequence of the method, full regularity of solution in two dimensions. The analysis leads to the study of regularity of a weak solution to a system that generalizes the Stokes system in two directions: the Laplace operator is replaced by a general linear second order elliptic operator and the pressure gradient is replaced by a linear first order operator, acting on the pressure.
Let w be an equality word of two binary non-periodic morphisms g, h : {a, b}∗ → ∆∗ with unique ov... more Let w be an equality word of two binary non-periodic morphisms g, h : {a, b}∗ → ∆∗ with unique overflows. It is known that if w contains at least 25 occurrences of each of the letters a and b, then it has to have one of the following special forms: up to the exchange of the letters a and b either w = (ab)a, or w = ab with gcd(i, j) = 1. We will generalize the result, justify this bound and prove that it can be lowered to nine occurrences of each of the letters a and b.
belong to C loc (Ω,R ). The system (1) has been extensively studied. S. Campanato in [2,3] proved... more belong to C loc (Ω,R ). The system (1) has been extensively studied. S. Campanato in [2,3] proved that (under suitable assumptions) Du ∈ L loc (Ω,R ) with λ < n, and u ∈ C 0,γ loc (Ω,R ) for some γ < 1 if n = 3, 4. If Ω has a smooth boundary and ai are differentiable and have controllable growth, then there is a positive ǫ such that u ∈ W 2,2+ǫ loc (Ω,R ) which implies that Du is Hölder continuous on Ω for n = 2 (see [8, 12, 13]). For this reason we will concentrate on the case
We prove the global existence of C 1,α -solutions to a system of nonlinear equations describing s... more We prove the global existence of C 1,α -solutions to a system of nonlinear equations describing steady planar motions of a certain class of non-Newtonian fluids including in particular variants of the power-law models. We study the Dirichlet problem. The nonlinear operator has a p-potential structure. If p>3/2 we construct global C 1,α -solutions up to the boundary, while for p>6/5 solutions with interior C 1,α -regularity are obtained. A proof of global higher regularity is outlined. Uniqueness of C 1,α -solutions within the class of weak solutions is also proved assuming the smallness of data.
We prove the existence of regular solution to a system of nonlinear equations describing the stea... more We prove the existence of regular solution to a system of nonlinear equations describing the steady motions of a certain class of non-Newtonian fluids in two dimen- sions. The equations are completed by requirement that all functions are periodic.
ABSTRACT We state a partial regularity result for steady flows of a class of incompressible fluid... more ABSTRACT We state a partial regularity result for steady flows of a class of incompressible fluids, their viscosity varies with the shear-rate and the pressure. Apart from partial regularity in three dimension we obtain, as a consequence of the method, full regularity of solution in two dimensions. The analysis leads to the study of regularity of a weak solution to a system that generalizes the Stokes system in two directions: the Laplace operator is replaced by a general linear second order elliptic operator and the pressure gradient is replaced by a linear first order operator, acting on the pressure.
Let w be an equality word of two binary non-periodic morphisms g, h : {a, b}∗ → ∆∗ with unique ov... more Let w be an equality word of two binary non-periodic morphisms g, h : {a, b}∗ → ∆∗ with unique overflows. It is known that if w contains at least 25 occurrences of each of the letters a and b, then it has to have one of the following special forms: up to the exchange of the letters a and b either w = (ab)a, or w = ab with gcd(i, j) = 1. We will generalize the result, justify this bound and prove that it can be lowered to nine occurrences of each of the letters a and b.
belong to C loc (Ω,R ). The system (1) has been extensively studied. S. Campanato in [2,3] proved... more belong to C loc (Ω,R ). The system (1) has been extensively studied. S. Campanato in [2,3] proved that (under suitable assumptions) Du ∈ L loc (Ω,R ) with λ < n, and u ∈ C 0,γ loc (Ω,R ) for some γ < 1 if n = 3, 4. If Ω has a smooth boundary and ai are differentiable and have controllable growth, then there is a positive ǫ such that u ∈ W 2,2+ǫ loc (Ω,R ) which implies that Du is Hölder continuous on Ω for n = 2 (see [8, 12, 13]). For this reason we will concentrate on the case
We prove the global existence of C 1,α -solutions to a system of nonlinear equations describing s... more We prove the global existence of C 1,α -solutions to a system of nonlinear equations describing steady planar motions of a certain class of non-Newtonian fluids including in particular variants of the power-law models. We study the Dirichlet problem. The nonlinear operator has a p-potential structure. If p>3/2 we construct global C 1,α -solutions up to the boundary, while for p>6/5 solutions with interior C 1,α -regularity are obtained. A proof of global higher regularity is outlined. Uniqueness of C 1,α -solutions within the class of weak solutions is also proved assuming the smallness of data.
We prove the existence of regular solution to a system of nonlinear equations describing the stea... more We prove the existence of regular solution to a system of nonlinear equations describing the steady motions of a certain class of non-Newtonian fluids in two dimen- sions. The equations are completed by requirement that all functions are periodic.
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