Related papers: Global regularity for some axisymmetric Euler flow…
In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{\omega^{r}}{r},\frac{\omega^{\theta}}{r})$, we get several Prodi-Serrin type…
We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids…
In this paper, we investigate regularity criterion for the solution of the nematic liquid crystal flows in dimension three and two. We prove the solution $(u,d)$ is smooth up to time $T$ provided that there exists a positive constant…
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+…
For one dimensional or multidimensional compressible Euler system of polytropic gases, it is well known that the smooth solution will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the…
We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. Explicit solutions to such a water wave problem are static configurations where the surface is an unduloid, that is, a…
This paper investigates the global dynamics of the Euler--Riesz system in three dimensions, focusing on the well-posedness and large-time behavior of solutions near equilibrium. The system generalizes classical interactions by incorporating…
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 solve here the so called division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that semilinear wave equations which can be written as systems involving quadratic derivative…
In this paper we will prove that the vorticity belongs to L1(0; T ; L2(\Omega)) for 3D incompressible Navier-Stokes equation with periodic initial-boundary value conditions, then the existence of a global smooth solution is obtained. Our…
In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the…
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first…
Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the…
We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the…
Supersonic flows for the two-dimensional (2D) steady full Euler system are studied. We construct a global non-isentropic rotational supersonic flow in a semi-infinite divergent duct. The flow satisfies the slip condition on the walls of the…
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…
We show that the homogeneous viscous Burgers equation $(\partial_t-\eta\Delta) u(t,x)+(u\cdot\nabla)u(t,x)=0,\ (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d$ $(d\ge 1, \eta>0)$ has a globally defined smooth solution if the initial condition…
We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…
In this paper we are concerned with the global existence of smooth solutions to the turbulent flow equations for compressible flows in $\mathbb{R}^3$. The global well-posedness is proved under the condition that the initial data are close…
We construct global weak solutions of the Euler equations in an infinite cylinder $\Pi=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for axisymmetric initial data without swirl when initial vorticity…