Related papers: Well-posedness of the hydrostatic Navier-Stokes eq…
We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy…
The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the…
We consider the Navier-Stokes equations in a bounded domain with periodic boundary conditions. Let $V=V(x,t)$ be the velocity of the fluid. The aim of this paper is to prove the bound $\|V(t)\|_{H^1}\le c$ for any $t\in\mathbb{R}_+$, where…
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we…
Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each…
Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in…
A constructive numerical approximation of the two-dimensional unsteady stochastic Navier-Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a parameter $\epsilon$. Space and time are…
The system of three dimensional Navier-Stokes equations is considered. We obtain some new local energy bounds that enable us to improve several $\epsilon$-regularity criteria. They key idea here is to view the `head pressure' as a signed…
We consider the Navier-Stokes system in a bounded domain with a smooth boundary. Given a sufficiently regular time-dependent global solution, we construct a finite-dimensional feedback control that is supported by a given open set and…
By using the continuous induction method, we prove that the initial value problem of the three dimensional Navier-Stokes equations is globally well-posed in $L^p(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ for any $3<p<\infty$. The proof is rather…
In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index $\sigma\in[3/2, 2],$ we obtain the local in time well-posedness in the space of Gevrey class $G^\sigma$ in the tangential…
We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato's proof for the Navier-Stokes equations is used,…
In a recent article, J.-Y. Chemin, I. Gallagher and M. Paicu obtained a class of large initial data generating a global smooth solution to the three dimensional, incompressible Navier-Stokes equations. This data varies slowly in the…
The NS equation is considered (in 2 & 3 dimensions) with a fixed forcing on large scale; the stationary states form a family of probability distributions on the fluid velocity fields depending on a parameter R (Reynolds number). It is…
This paper concerns an inverse problem for the initial boundary value problem of the two-dimensional Navier-Stokes system defined in a bounded simply connected domain with slip, vorticity boundary conditions, and a global vorticity…
We consider the stationary problem for the quasi-geostrophic equation on the whole plane and investigate its well-posedness and ill-posedness. In[Fujii, Ann. PDE 10, 10 (2024)], it was shown that the two-dimensional stationary…
We consider the motion of compressible Navier-Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are non zero. This kind of pressure model is largely employed in various…
We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on…
This paper is devoted to studying the well-posedness, (conditional) conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell equations (NSM) under the Hall effect in two and three…
For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near…