PUB - Publikationen an der Universität Bielefeld

In my thesis we discuss the well-posedness and long-time behaviours of stochastic equations from fluid dynamics. More precisely, we consider the following topics:
1. We study the well-posedness of both deterministic and anisotropic 2D Navier–Stokes equations. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space \tilde {H}^*0,1*. For the stochastic case, we obtain the existence of martingale solutions and pathwise uniqueness of the solutions, which imply the existence of the probabilistically strong solution to this system by the Yamada–Watanabe Theorem.
2. We also study stationary solutions and ergodicity of both anisotropic Navier–Stokes equations and Euler equations with positive damping terms on both the torus T^*2* and the whole space ℝ^*2*. We first show the existence of (H^*1* -valued) martingale solutions for both the 2D anisotropic Navier–Stokes equations and Euler equations. Then we prove the existence of H^*1*-valued stationary martingale solutions for the equations with positive damping terms on both T^*2* and ℝ^*2*, while previous work in the literature focused on the stochastic Euler equations on T^*2*.
Finally, for the case of anisotropic Navier–Stokes equations with a positive damping term, we prove the uniqueness of the invariant measure when the noise term is small enough with respect to the damping term by the coupling method. Moreover, the convergence to the (unique) invariant measure is proved to be exponentially fast.
3. We show that there exists a white noise stationary solution of the modified Surface Quasi-Geostrophic equations on ℝ^*2*, i.e. there exists a stationary solution of the modified Surface Quasi-Geostrophic equations with the marginal distribution equal to space white noise in ℝ^*2*.
4. We investigate the stochastic 3D Navier–Stokes equations perturbed by Gaussian noise of convolution type by transformation to a Random PDE. Instead of obtaining global well-posedness results when the initial data are small enough, we focus on the mild solution in some time-space Sobolev space. We prove that for ℙ-a.e. path ω, there exists a mild solution in the time interval [0, T∗(u_*0*, ω)], where u_*0* is the initial condition.