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In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but…
Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The…
We study the long time behavior of isentropic compressible Euler equations with linear damping driven by a white-in-time noise, on a one-dimensional torus. We prove the existence of a statistically stationary solution in the class of weak…
This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness…
We are concerned with the three dimensional incompressible Navier--Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in $L^{2}$ we establish existence of infinitely…
We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear…
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…
The present work is devoted to introduce the backward Euler based modular time filter method for MHD flow. The proposed method improves the accuracy of the solution without a significant change in the complexity of the system. Since time…
Existence of stationary point vortices solution to the damped and stochastically driven Euler's equation on the two dimensional torus is proved, by taking limits of solutions with finitely many vortices. A central limit scaling is used to…
In this paper we establish local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions. An analogous result is…
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence…
We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato…
This article is devoted to stationary solutions of Euler's equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the…
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and…
In this paper, we are concerned with the global existence and optimal rates of strong solutions for three-dimensional compressible viscoelastic flows. We prove the global existence of the strong solutions by the standard energy method under…
This paper investigates projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We…
In this paper, we find various analytic (1+3)D solutions to relativistic ideal hydrodynamic equations based on embedding of known low-dimensional scaling solutions. We first study a class of flows with 2D Hubble Embedding, for which a…
In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid.…
One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that coarse scale averages of the velocity carry the fine scale…
In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the…