Related papers: Vorticity convergence from Boltzmann to 2D incompr…
In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that…
We present explicit expressions of the helicity conservation in nematic liquid crystal flows, for both the Ericksen-Leslie and Landau-de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation…
Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a…
We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. The domain is the exterior of a regular lattice of rigid particles. We study the simultaneous limit of vanishing particle size and…
We consider the hydrodynamics of relativistic conformal field theories at finite temperature. We show that the limit of slow motions of the ideal hydrodynamics leads to the non-relativistic incompressible Euler equation. For viscous…
In this article we investigate the two-dimensional incompressible rotating and stratified, just rotating, just stratified Euler equations with each other and with the normal Euler equations with the self-similar Ansatz. There are analytic…
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop…
We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time,…
We establish the existence of global weak solutions of the 2D incompressible Euler equation, for a large class of non-smooth open sets. These open sets are the complements (in a simply connected domain) of a finite number of connected…
We consider stochastic Navier-Stokes equations in a 2D-bounded domain with the Navier with friction boundary condition. We establish the existence and the uniqueness of the solutions and study the vanishing viscosity limit. More precisely,…
We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite…
This article is devoted to incompressible Euler equations (or to Navier-Stokes equations in the vanishing viscosity limit). It describes the propagation of quasi-singularities. The underlying phenomena are consistent with the notion of a…
In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t)…
A comparison theorem is proved for a pair of solutions that satisfy in a weak sense opposite differential inequalities with nonlinearity of the form $f (u)$ with $f$ belonging to the class $L^p_{loc}$. The solutions are assumed to have…
We consider an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$, each one of vorticity mass and…
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^\alpha$…
In this article our goal is to study the singular limits for a scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach number $=\epsilon^m $, Rossby number $=\epsilon $ and Froude number $=\epsilon^n…
We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov…
Currently, Eulerian flow solvers are very efficient in accurately resolving flow structures near solid boundaries. On the other hand, they tend to be diffusive and to dampen high-intensity vortical structures after a short distance away…
We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport…