Study of a chemo-repulsion model with quadratic production. Part I: Analysis of the continuous problem and time-discrete numerical schemes

Abstract

We consider a chemo-repulsion model with quadratic production in a bounded domain. Firstly, we obtain global in time weak solutions, and give a regularity criterion (which is satisfied for 1D and 2D domains) to deduce uniqueness and global regularity. After, we study two cell-conservative and unconditionally energy-stable first-order time schemes: a (nonlinear and positive) Backward Euler scheme and a linearized coupled version, proving solvability, convergence towards weak solutions and error estimates. In particular, the linear scheme does not preserve positivity and the uniqueness of the nonlinear scheme is proved assuming small time step with respect to a strong norm of the discrete solution. This hypothesis is reduced to small time step in nD domains (n ≤ 2) where global in time strong estimates are proved. Finally, we show the behavior of the schemes through some numerical simulations.

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

“…where v ε = v( z ε ) and, in general, we denote a + := max{a, 0}. Then, (u ε , z ε ) is a solution of (13) iff (u ε , z ε ) is a fixed point of the operator R defined in (16). Let us check every hypotheses of Leray-Schauder Theorem:…”

“…In the case of linear (p = 1) and quadratic (p = 2) production terms, the problem (2) is wellposed (see [7,13] respectively) in the following sense: there exist global in time weak solutions (based on an energy inequality) and, for 2D domains, there exists a unique global in time strong solution. However, as far as we know, there are not works studying problem (2) with production u p , with 1 < p < 2.…”

“…An energy-stable finite volume scheme for a Keller-Segel model with an additional cross-diffusion term has been studied in [6]. In [13,14], unconditionally energy stable time-discrete numerical schemes and fully discrete FE schemes for a chemo-repulsion model with quadratic production have been analyzed. In [15], the authors studied unconditionally energy stable fully discrete FE schemes for a chemo-repulsion model with linear production.…”

On a bi-dimensional chemo-repulsion model with nonlinear production

“…However, as far as we know, in the case of chemo-repulsion model with nonlinear signal production (1.1)-(1.2) (in particular, the quadratic production) the literature is very scarce. We only known the results of [7,8,9]; in [7,8] the authors prove the existence of global weak solutions for both two and three dimensions of (1.1)-(1.2) in the quadratic case, with f ≡ 0, and global in time strong regularity of the model assuming a regularity criteria, which is satisfied in 2D domains. They also develop some numerical schemes to approximate weak solutions of (1.1)- (1.2).…”