Related papers: Removing Type II singularities off the axis for th…

We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's…

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that $u_0 \in H^{2.5+\delta }$ is such that $\mathrm{curl}\,u_0 \in H^{2+\delta }$ in an arbitrarily small neighborhood of…

We consider the Schr\"odinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up…

Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in…

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D…

In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the…

We study the vanishing viscosity limit for the three-dimensional incompressible Navier-Stokes equations in terms of the relative vorticity in the setting of axisymmetric velocity fields without swirl. We show that the weak convergence of…

In the note, a certain scenario of potential Type II blowups of axisymmetric solutions to the Navier-Stokes equations is considered. The main tool of the treatment of such blowups is the corresponding Euler scaling.

We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady…

In this paper, we mainly investigate the tridimensional incompressible axisymmetric Euler equations without swirl in the whole space. Specifically, we prove the global existence of weak solutions if the swirl component of initial vorticity…

We study mean field equations with singular sources on a compact Riemann surface with boundary $(\Sigma,g)$, subject to homogeneous Neumann boundary conditions: \[ -\Delta_g v = \rho\left( \frac{V e^{v}}{\int_\Sigma V e^{v}\, d v_g} -…

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb…

We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual…

Using group theory arguments, we demonstrate that, unlike in homogeneous media, no symmetric vortices of arbitrary order can be generated in two-dimensional (2D) nonlinear systems possessing a discrete-point symmetry. The only condition…

It is shown that a local-in-time strong solution $u$ to the 3D Navier-Stokes equations remains regular on an interval $(0,T)$ provided a smallness $\epsilon_0$-condition on $u$ in a lower time-restricted local Morrey space is stipulated;…

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular…

We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$…

The global existence of weak solutions for the three-dimensional axisymmetric Euler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$…

For the 3d cubic nonlinear Schr\"odinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time…

We consider the Cauchy problem for the energy critical heat equation $$ u_t = \Delta u + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$.…