Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space \nR^3 and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Since the main purpose of this line of research is to introduce a reliable and stable inviscid numerical regularization of the underlying model we, in particular, show that the solutions of the Voigt regularized system converge, as the regularization parameter \alpha\maps0, to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization. This type of regularization, and the corresponding results, are valid for, and can also be applied to, a wide class of hydrodynamic models

Regularisation and Long-Time Behaviour of Random Systems

Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and uniqueness of a strong solution to the TMHD equations, prove that smooth data lead to smooth solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the solutions to the tamed equations converge to a weak solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a unique strong solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations

Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth

In this paper, we study an initial boundary value problem of the Cahn-Hilliard-Darcy system with a non-autonomous mass source term $S$ that models tumor growth. We first prove the existence of global weak solutions as well as the existence of unique local strong solutions in both 2D and 3D. Then we investigate the qualitative behavior of solutions in details when the spatial dimension is two. More precisely, we prove that the strong solution exists globally and it defines a closed dynamical process. Then we establish the existence of a minimal pullback attractor for translated bounded mass source $S$. Finally, when $S$ is assumed to be asymptotically autonomous, we demonstrate that any global weak/strong solution converges to a single steady state as $t\to+\infty$. An estimate on the convergence rate is also given

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.Comment: 37 pages; references added, minor typos corrected, minor changes to revise for publicatio