Asymptotic expansions in time for rotating incompressible viscous fluids

We study the three-dimensional Navier--Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray-Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincar\'e waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds

On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program. John Simon Guggenheim Memorial Foundation

Global regularity for a rapidly rotating constrained convection model of tall columnar structure with weak dissipation

We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.Einstein Stiftung/Foundation - Berlin and John Simon Guggenheim Memorial Foundation

Global strong solutions for the three-dimensional Hasegawa-Mima model with partial dissipation

We study the three-dimensional Hasegawa-Mima model of turbulent magnetized plasma with horizontal viscous terms and a weak vertical dissipative term. In particular, we establish the global existence and uniqueness of strong solutions for this model

Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data

Funder: Einstein Stiftung Berlin; doi: http://dx.doi.org/10.13039/501100006188Funder: John Simon Guggenheim Memorial Foundation; doi: http://dx.doi.org/10.13039/100005851This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $\mathcal O(\varepsilon)$, as $\varepsilon \rightarrow 0^+$, where $\varepsilon$ represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows

Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws.

We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order one-third in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary. This extends various recent results of the authors

Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number

Funder: Einstein Stiftung Berlin; doi: http://dx.doi.org/10.13039/501100006188We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model