Related papers: On the time of existence of solutions of the Euler…
The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg…
We consider the Euler-Korteweg system with space periodic boundary conditions $ x \in \mathbb T^d $. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity…
The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of…
Global existence of solutions to the compressible Navier-Stokes-Korteweg system around a constant state is studied. This system describes liquid-vapor two phase flow with phase transition as diffuse interface model. In previous works they…
The initial-boundary value problem for the Schr\"odinger-Korteweg-de Vries system is considered on the left and right half-line for a wide class of initial-boundary data, including the energy regularity $H^1(\R^{\pm})\times H^1(\R^{\pm})$…
We are concerned with the global solution of the compressible Euler-Korteweg equations in $\mathbb{R}^{3}$. In the case of zero sound speed $P'(\rho^{\ast})=0$, it is found that the perturbation problem of irrotational fluids could be…
The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws…
In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the…
This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We will prove the existence…
This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We will prove the existence…
For an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative…
Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In…
A rotating stationary solution of the vacuum Einstein equations with a cosmological constant is exhibited which reduces to de Sitter's interior cosmological solution when the angular momentum goes to zero. This solution is locally…
We consider a general Euler-Korteweg-Poisson system in $R^3$, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as…
We prove the uniqueness and finite-time existence of bounded-vorticity solutions to the 2D Euler equations having velocity growing slower than the square root of the distance from the origin, obtaining global existence for more slowly…
In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for…
We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by…
We are concerned with the decay of long time solutions of the initial value problem associated with the Schr\"odinger-Korteweg-de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the…