Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes

We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll} \partial_t u - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ \Omega,\ t>0,\\ \partial_t v - \Delta v + v = u^p \ \ \mbox{in}\ \Omega,\ t>0, \end{array} \right. \end{equation} in a bounded domain $\Omega\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the existence of solutions of this problem. Moreover, we propose three fully discrete Finite Element (FE) nonlinear approximations, where the first one is defined in the variables $(u,v)$, and the second and third ones by introducing ${\boldsymbol\sigma}=\nabla v$ as an auxiliary variable. We prove some unconditional properties such as mass-conservation, energy-stability and solvability of the schemes. Finally, we compare the behavior of the schemes throughout several numerical simulations and give some conclusions.Comment: arXiv admin note: substantial text overlap with arXiv:1807.0111

Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions