Related papers: Lower bounds for possible singular solutions for t…
The limit resonant equation of the 3D rotating Navier-Stokes equations is obtained by taking large rotation limit. This equation has a nonlinear term with restricted interactions between Fourier modes, and thus it enjoys better regularity…
We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $\omega$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times…
We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on $\R^{2}$. In our previous work we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so…
We study steady vortex sheet solutions of the Navier-Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a…
Assuming that ${T}$ is a potential blow up time for the Navier-Stokes system in half-space, we show that $L_{3}$-norm of the velocity field goes to $\infty$ as time t approaches $T$.
The incompressible Navier-Stokes equations and static Euler equations are considered. We find that there exist infinite non-trivial regular solutions of incompressible static Euler equations with given boundary conditions. Moreover there…
We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on…
In this paper, we study the potential singular points of interior and boundary suitable weak solutions to the 3D Navier--Stokes equations. It is shown that upper box dimension of interior singular points and boundary singular points are…
Assuming some \dot H^{1/2} - initial data lead to a singularity for the 3d Navier-Stokes equations, we show that there are also initial data with the minimal \dot H^{1/2} - norm which will produce a singularity.
The object of the present paper is to show the existence and the uniqueness of a reproductive strong solution of the Navier-Stokes equations, i.e. the solution $\boldsymbol{u} $ belongs to $\text{}\mathbf{L}% ^{\infty}(0,T;V) \cap…
The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic…
The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the…
We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be…
In this paper we first show a blow-up criterion for solutions to the Navier-Stokes equations with a time-independent force by using the profile decomposition method. Based on the orthogonal properties related to the profiles, we give some…
We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is…
We show that smallness of $\dot B^{-1}_{\infty,\infty}$ norm of solution to $d$-dimensional ($d\ge 3$) incompressible Navier-Stokes prevents blowups.
This paper is concerned with the localized behaviors of the solution $u$ to the Navier-Stokes equations near the potential singular points. We establish the concentration rate for the $L^{p,\infty}$ norm of $u$ with $3\leq p\leq\infty$.…
In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schr\"odinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial data is radially symmetric in the…