Related papers: Weak vorticity formulation of 2D Euler equations w…

This work considers the two-dimensional Allen-Cahn equation $$ \partial_t u = \frac{1}{2}\Delta u + \mathfrak{m}\, u -u^3\;, \quad u(0,x)= \eta (x)\;, \qquad \forall (t,x) \in [0, \infty) \times \mathbb{R}^{2} \;, $$ where the initial…

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in…

We consider the limit $\alpha\to0$ for the $\alpha$-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in $L^p$, we prove the existence of a global solution and we…

In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $1<\gamma\leq 3$, by constructing some new control functions ingeniously, we obtain the lower bounds estimates…

We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $L^2$ weak limit of Leray--Hopf weak solutions of the Navier-Stokes equations on any…

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In…

We consider the two-dimensional incompressible Euler equation \[\begin{cases} \partial_t \omega + u\cdot \nabla \omega=0 \\ \omega(0,x)=\omega_0(x). \end{cases}\] We are interested in the cases when the initial vorticity has the form…

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the…

We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity…

In this work, we study the initial boundary value problem for a non-strictly hyperbolic $2\times2$ system of equations in the quarter plane $x>0,t>0$ which is derived from Eulerian droplet model for air particle flow for velocity and volume…

We study the evolution of solutions to the 2D Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely $L^p$ integrable for some…

We consider the Cauchy problem for the 3D incompressible axisymmetric swirl-free Euler equations. The convex integration method developed by De Lellis and Sz\'ekelyhidi rules out the possibility that the Euler equations admit unique…

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based…

We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $\rho_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, \rho_0,…

In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential…

In this paper, we consider 2D incompressible Euler equations in an unbounded domain with a free surface and a fixed bottom at finite depth. The fluid motion is under the influence of gravity and surface tension. We construct initial data…

We compute the generation of vorticity from velocity dispersion in the dark matter fluid. For dark matter at zero temperature Helmholtz's theorem dictates that no vorticity is generated and we therefore allow the dark matter fluid to have a…

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and…

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity…

We consider the 2D incompressible Euler equation on a corner domain $\Omega$ with angle $\nu\pi$ with $\frac{1}{2}<\nu<1$. We prove that if the initial vorticity $\omega_0 \in L^{1}(\Omega)\cap L^{\infty}(\Omega)$ and if $\omega_0$ is…