Stochastic Euler Equations Research Articles

Theory of Weak Asymptotic Autonomy of Pullback Stochastic Weak Attractors and Its Applications to 2D Stochastic Euler Equations Driven by Multiplicative Noise

Theory of Weak Asymptotic Autonomy of Pullback Stochastic Weak Attractors and Its Applications to 2D Stochastic Euler Equations Driven by Multiplicative Noise

Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces

Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces

A New Approach to Estimation of Stochastic Euler Equations

A New Approach to Estimation of Stochastic Euler Equations

On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow

We introduce a concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probabilistic sense i.e., the underlying probability space and the driving Wiener process are intrinsic parts of the solution. We first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise and then demonstrate the pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, á la Prodi (Ann Mat Pura Appl 48:173–182, 1959) and Serrin (in: Nonlinear problems, University of Wisconsin Press, Madison, Wisconsin, pp 69–98, 1963), for the uniqueness of weak martingale solutions to the stochastic Navier–Stokes system in the class of finite energy weak martingale solutions.

Dissipative Solutions to the Stochastic Euler Equations

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier–Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. As a main novelty, our solutions satisfy a form of the energy inequality which gives rise to a weak–strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.

Existence and Uniqueness of the Local Smooth Solution to 3D Stochastic MHD Equations without Diffusion.

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in . We first transform the system into a random system via a simple change of variable and borrow the result obtained for classical magneto-hydrodynamic equations, then we show that this random transformed system is measurable with respect to the stochastic element. Finally we extend the solution to the maximality solution. Due to the coupled construction of this system, we need more elaborate and complicated estimates with respect to stochastic Euler equation.

Local existence for the stochastic Euler equations with Lévy noise

Local existence for the stochastic Euler equations with Lévy noise

$$\rho $$ ρ -White noise solution to 2D stochastic Euler equations

A stochastic version of 2D Euler equations with transport type noise in the vorticity is considered, in the framework of Albeverio--Cruzeiro theory [1] where the equation is considered with random initial conditions related to the so called enstrophy measure. The equation is studied by an approximation scheme based on random point vortices. Stochastic processes solving the Euler equations are constructed and their density with respect to the enstrophy measure is proved to satisfy a continuity equation in weak form. Relevant in comparison with the case without noise is the fact that here we prove a gradient type estimate for the density. Although we cannot prove uniqueness for the continuity equation, we discuss how the gradient type estimate may be related to this open problem.

Euler-Lagrangian approach to 3D stochastic Euler equations

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation.

Stochastic Euler equations of fluid dynamics with Lévy noise

Abstract : In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lvy noise in two and three dimensions. The existence of a unique maximal solution is also proved.

Existence and Uniqueness for Stochastic 2D Euler Flows with Bounded Vorticity

Strong existence and pathwise uniqueness of solutions with $L^{\infty}$-vorticity of 2D stochastic Euler equations is proved. The noise is multiplicative and involves first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. Stability under regularization is also proved.

Measure valued solutions to the stochastic Euler equations in $$\mathbb {R}^d$$ R d

We establish the existence of measure-valued solutions of the stochastic Euler equations in \(\mathbb {R}^d, \, d \ge 2.\) Our definition of a measure-valued solution is based on that for the deterministic Euler equations. Our first result is the existence over a given probability space in the spirit of Bensoussan and Temam (J Funct Anal 13:195–222, 1973). The main tool is measurable selection established in Stroock and Varadhan (Multidimensional diffusion processes, 1997). As a necessary step for this result, we also present a new existence result on the stochastic Navier–Stokes equations. Our second result is the existence of a martingale measure-valued solution. The main tool is the generalized Skorokhod representation theorem due to Jakubowski (Theory Probab Appl, 42:167–174, 1998).

Inviscid limit for 2D stochastic Navier–Stokes equations

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 prove that solutions of stochastic Navier–Stokes equations converge, as the viscosity goes to zero, to solutions of the corresponding stochastic Euler equations.

On a 2D stochastic Euler equation of transport type: Existence and geometric formulation

We prove weak existence of Euler equation (or Navier–Stokes equation) perturbed by a multiplicative noise on bounded domains of ℝ2 with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are H1 regular. The equations are of transport type.

Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations

We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box, for any power of the dissipation term.

Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise

We establish the local existence of pathwise solutions for the stochastic Euler equations in a three-dimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Finally, we show that, in the three-dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large.

The Evolution of Loss Aversion Coefficient in Energy Futures Market Considering Investor Heterogeneity

The energy futures market has become the center of global energy market, so, problems in the energy futures market have profound impact on many aspects of the macroeconomic. In this paper, with the premise of the existence of investor heterogeneity, we use stochastic Euler equation and prospect theory to establish a mathematical model of price fluctuations which varies with the time change, and the model is testable for generalized method of moment (GMM) model. According to the GMM estimation results, in the one-year period, when we set a certain time point to be the initial point, with the passage of time, loss aversion coefficient will gradually increase, while it will reduce with the increase of the discount factor. We also verify that the loss aversion coefficient of investors who response to market information rapidly is relatively small. But in the long term, the evolution of loss aversion coefficient needs further research.

Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.

On the statistical mechanics of the 2D stochastic Euler equation

The dynamics of vortices and large scale structures is qualitatively very different in two dimensional flows compared to its three dimensional counterparts, due to the presence of multiple integrals of motion. These are believed to be responsible for a variety of phenomena observed in Euler flow such as the formation of large scale coherent structures, the existence of meta-stable states and random abrupt changes in the topology of the flow. In this paper we study stochastic dynamics of the finite dimensional approximation of the 2D Euler flow based on Lie algebra su(N) which preserves all integrals of motion. In particular, we exploit rich algebraic structure responsible for the existence of Euler's conservation laws to calculate the invariant measures and explore their properties and also study the approach to equilibrium. Unexpectedly, we find deep connections between equilibrium measures of finite dimensional su(N) truncations of the stochastic Euler equations and random matrix models.Our work can be regarded as a preparation for addressing the questions of large scale structures, meta-stability and the dynamics of random transitions between different flow topologies in stochastic 2D Euler flows.

Existence of a local smooth solution in probability to the stochastic Euler equations in [formula omitted

We establish the existence of a local smooth solution of the stochastic Euler equations in R 3 . Probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise. There are numerous works on the stochastic Euler equations in a two-dimensional domain. Even for the deterministic Euler equations in a three-dimensional domain, the only results are concerned with the local existence of smooth solutions. Our goal is to extend such local existence result to the stochastic equations.