Related papers: Gradient estimates for Stokes and Navier-Stokes sy…
We study interior regularity of solutions of a generalized stationary Stokes problem in the plane. The main, elliptic part of the problem is given in the form div(A(Du)), where D is the symmetric part of the gradient. The model case is…
We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $\Omega$ under the condition…
We consider suitable weak solutions of the incompressible Navier--Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of…
We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in…
We consider the question of existence of weak solutions for the fully inhomogeneous, stationary generalized Navier-Stokes equations for homogeneous, shear-thinning fluids. For a shear rate exponent $p \in \big(\tfrac{2d}{d+1}, 2\big)$,…
We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for…
This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and $L^\infty$ estimates for the pressure as…
We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $\Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $\a \in L^2(\partial\Omega)$ satisfies ${1\over…
We use the scale $B^s_{\tau}(L_\tau(\Omega))$, $1/\tau=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity…
In this paper, we consider the fractional Navier-Stokes equations. We extend a previous non-uniqueness result due to Cheskidov and Luo, found in [5], from Navier-Stokes to the fractional case, and from $L^1$-in-time, $W^{1,q}$-in-space…
We prove well-posedness in reflexive Sobolev spaces of weak solutions to the stationary Stokes problem with Navier slip boundary condition over bounded domains $\Omega$ of $\mathbb{R}^n$ of class $W^{2-1/s}_s$, $s>n$. Since such domains are…
We prove existence of weak solutions for the fully inhomogeneous, stationary generalized Navier-Stokes equations for shear-thinning fluids. Our proof is based on the theory of pseudomonotone operators and the Lipschitz truncation method,…
We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order…
The aim of this work is to analyze the finite element approximation of the two-dimensional stationary Navier-Stokes equations with non-smooth Dirichlet boundary data. The discrete approximation is obtained by considering the Navier-Stokes…
We aim at proving existence of weak solutions to the stationary compressible Navier-Stokes system coupled with the Allen-Cahn equation. The model is studied in a bounded three dimensional domain with slip boundary conditions for the…
Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg, as a corollary, under suitable assumptions of local character on the initial data, we prove a…
In this paper, we establish the strong($H^1$) well-posedness of the two dimensional stochastic Navier-Stokes equation with multiplicative noise on moving domains. Due to the nonlocality effect, this equation exhibits a ``piecewise"…
We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear…
We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition. Our result holds provided certain condition on the shape of the boundary around…
We prove, with a more geometric approach, that the solutions to the Navier-Stokes equations are regular up to a set of Hausdorff dimension 1. The main tool for the proof is a new compactness lemma and the monotonicity property of harmonic…