Vorticity-Velocity formulation for the stationary Navier-Stokes equations: The three-dimensional case

AbstractIn this article, we propose a mixed method for the vorticity-velocity formulation of the stationary Stokes and Navier-Stokes equations in space dimension three, the unknowns being the vorticity and the velocity of the fluid

Pullback attractors for a non-autonomous homogeneous two-phase flow model

AbstractThis article studies the pullback asymptotic behavior of solutions for a non-autonomous homogeneous two-phase flow model in a two-dimensional domain. We prove the existence of pullback attractors AV in V (the velocity has the H1-regularity) and AY in Y (the velocity has the L2-regularity). Then we verify the regularity of the pullback attractors by proving that AV=AY, which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data. The method used in this article is similar to the one used in Zhao and Zhou (2007) [42] in the case of the non-autonomous incompressible non-Newtonian fluid in a two-dimensional domain. Let us mention that the nonlinearity involved in the model considered in this article is stronger than the one in the two-dimensional non-Newtonian flow studied in Zhao and Zhou (2007) [42]

Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523

Existence and regularity of strong solutions to a nonhomogeneous Kelvin-Voigt-Cahn-Hilliard system

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with the convective Cahn-Hilliard equation. This system describes the evolution of an incompressible isothermal Newtonian mixture of binary fluids. In this article we investigate a variant of this model, which consists of the nonhomogeneous Kelvin-Voigt equations coupled with the Cahn-Hilliard equations. We prove the existence of global weak and strong solutions in a two and three dimensions. Furthermore, we prove some regularity results for the strong solutions and show their uniqueness in both two or three-dimensional bounded domains. Lastly, we also solve a problem of uniqueness of regular solutions that was left open by Giorgini and Temam (2020) [5]