Related papers: A weighted setting for the stationary Navier Stoke…
Our aim is to prove Liouville type theorems for the three dimensional steady-state Navier-Stokes equations provided the velocity field belongs to some Lorentz spaces. The corresponding statement contains several known results as a…
In this paper we propose new method for proving of global solutions for 3D Navier-Stokes equations. This complies an application to the Clay Institute Millennium Prize Navier Stokes Problem. The proposed method can be applied for…
Higher-order estimates in weighted Sobolev spaces for solutions to a singular elliptic equation for the stream function in an axially symmetric cylinder are provided. These estimates are essential for investigating the existence of axially…
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type…
We study the existence of a strong solution to the initial value problem for the incompressible Navier-Stokes equations in the whole space. Our investigation shows that a ``suitable'' weak solution to the problem becomes a strong one…
We consider the Cauchy problem to the axisymmetric Navier-Stokes equations. To prove an existence of global regular solutions we examine the Navier-Stokes equations near the axis of symmetry and far from it separately. We derive only a…
We consider the 2D stochastic Navier-Stokes equations driven by noise that has the regularity of space-time white noise but doesn't exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at…
This paper establishes global weighted Calder\'on-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the…
We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl}…
The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends…
Chae and Wolf recently constructed discretely self-similar solutions to the Navier-Stokes equations for any discretely self similar data in $L^2_{\mathrm{loc}}$. Their solutions are in the class of local Leray solutions with projected…
In this paper, we consider the three-dimensional inhomogeneous Navier-Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when $\|\nabla…
We study the linear elasticity system subject to singular forces. We show existence and uniqueness of solutions in two frameworks: weighted Sobolev spaces, where the weight belongs to the Muckenhoupt class $A_2$; and standard Sobolev spaces…
This paper concerns the existence of global weak solutions \`a la Leray for compressible Navier-Stokes equations with a pressure law that depends on the density and on time and space variables $t$ and $x$. The assumptions on the pressure…
The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the…
The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including…
For the stochastic Navier-Stokes equations with a multiplicative white noise on ${\mathbb T}^{3}$, we prove that there exists a unique strong solution locally in time when the initial datum belongs to $L^p(\Omega; L^p)$ for $p>3$.
In this paper, we discuss the existence of local strong solutions for the multivalued version of three-dimensional nonstationary Navier-Stokes equation in Banach spaces. Also, we considered a more general inclusion problem and studied the…
We prove that there exists a nontrivial finite energy periodic stationary weak solution to the 3D Navier-Stokes equations (NSE). The construction relies on a convex integration scheme utilizing new stationary building blocks designed…
In this paper, we show the existence and uniqueness of a strong solution to stochastic 3D tamed Navier-Stokes equations driven by multiplicative Levy noise with periodic boundary conditions. Then we establish the large deviation principles…