We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)+μ(u-u^2),\quad x\in Ω, t>0, \disp{v_t-Δv=-uv },\quad x\in Ω, t>0, %\disp{τw_t+δw=u },\quad %x\in Ω, t>0, \disp{(\nabla D(u)-χu\cdot \nabla v)\cdot ν=\frac{\partial v}{\partialν}=0},\quad x\in \partialΩ, t>0, \disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in Ω\end{array}\right. $$ on a bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partialΩ, χ$ and $μ$ are positive constants. Besides appropriate smoothness assumptions, in this paper it is only required that $D(u)\geq C_{D}(u+1)^{m-1}$ for all $u\geq 0$ with some $C_{D} > 0$ and some $$ m>\left\{\begin{array}{ll} 1-\fracμ{χ[1+λ_{0}\|v_0\|_{L^\infty(Ω)}2^{3}]}~~\mbox{if}~~ N\leq2, % >1+\frac{(N+2-2r)^+}{N+2}~~~~~~\mbox{if}~~ % \frac{N+2}{2}\geq r\geq\frac{N+2}{N}, 1~~~~~~\mbox{if}~~ N\geq3, \end{array}\right. $$ then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where $λ_{0}$ is a positive constant which is corresponding to the maximal sobolev regularity. The results of this paper extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who proved the possibility of boundness of weak solutions, in the case $m>1$ and $N=3$.
This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=Δu-\nabla\cdot(u\nabla v), v_t=Δv-uv, \end{array}\right. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $Ω\subset\mathbb{R}^n~(n\geq4)$ with smooth boundary. It is proved that if initial data satisfy $u_0\in C^0(\overlineΩ)$ and $v_0\in W^{1,q}(Ω)$ for some $q>n$, the model possesses at least one global renormalized solution.
Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these non-linear parameter functions in a chemotaxis model with volume filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.
We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant. This paper partially improves some known results in the literature and moreover solves an open question. The research is also complemented with numerical simulations in bi-dimensional domains.
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.
We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production $g$ for the chemorepellent and sublinear rate $f$ for the chemoattractant: \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - χ\nabla \cdot (u \nabla v)+ξ\nabla \cdot (u \nabla w) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-f(u)v & \text{ in } Ω\times (0,T_{max}),\\ 0= Δw - δw + g(u)& \text{ in } Ω\times (0,T_{max}). %u(x,0)=u_0(x), \; v(x,0)=v_0(x) & x \in \barΩ. \end{cases} \end{equation} In this problem, $Ω$ is a bounded and smooth domain of $\R^n$, for $n\geq…
This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates global-in-time solution. In view of our theorem one sees that one-dimensional Keller- Segel system is essentially different than its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new func…
We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \begin{align*} \left\{\begin{array}{c@{\,}l@{\quad}l@{\,}c} u_{t}&=Δu-\nabla\!\cdot(u\nabla v),\ &x\in\mathbb{R}^n,& t>0,\\ 0 &=Δv+u+f(x),\ &x\in\mathbb{R}^n,& t>0,\\ u(x,0)&=u_{0}(x),\ &x\in\mathbb{R}^n,& \end{array}\right. \end{align*} with prototypical external signal production \begin{align*} f(x):=\begin{cases} f_0\vert x\vert^{-α},&\text{ if }\vert x\vert \leq R-ρ,\\ 0,&\text{ if } \vert x\vert\geq R+ρ,\\ \end{cases} \end{align*} for $R\in(0,1)$ and $ρ\in\left(0,\frac{R}{2}\right)…
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in Ω,\ t>0, \\[1mm] 0 = Δv - μ+ u, &x\in Ω,\ t>0, \end{cases} \end{align*} where $Ω:= B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $χ>0$, $p,q\geq1$, $μ:= \frac 1{|Ω|} \int_Ωu_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;20…
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = Δv - μ+ u \end{cases}\end{equation*} under no-flux boundary conditions in balls $Ω\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overlineΩ)$, where $χ>0$ and $μ:=\frac{1}{|Ω|}\int_Ωu_0$. Bellomo--Winkler (Comm.\ Partial Differential Equa…
This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + λu - μu^κ, & x\inΩ,\ t>0, \\[1mm] 0=Δv - \overline{M_f}(t) + f(u), & x\inΩ,\ t>0, \end{cases} \end{equation*} where $λ>0$, $μ>0$, $κ>1$ and $\overline{M_f}(t):=\frac{1}{|Ω|}\int_Ω f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{α-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), x\in Ω, t>0,\\ τv_t=Δv- v +u,\quad x\in Ω, t>0,\\ w_t=- vw,\quad x\in Ω, t>0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, where $τ\in\{0,1\}$ and $χ$, $ξ$ and $μ$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result…
We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global…
The paper should be viewed as complement of an earlier result in [8]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form 1/u all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [8] deals with the Jager-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [8], being a main step of th…
As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in microdevices, we develop a framework for constructing approximate analytical solutions for the location, speed and cellular densities for cell chemotaxis waves in heterogeneous fields of chemoattractant from the underlying partial differential equation models. In particular, such chemotactic waves are not in general translationally invariant travelling waves, but possess a spatial variation that evolves in time, and may even may oscillate back and forth in time, according to the details of the chemot…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big),\\[] 0=Δv+αu-βv,\\[] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$ with smooth boundary $\partial Ω$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. In the case that $m=1$ and $p=q=2$, when $χα-ξγ<0$ and $β=δ$, Tao-Wang (Math. Models Methods Appl. Sci.; 2013; 23; 1-36) proved that global bounded classical solutions toward th…
In this paper we study a general class of hybrid mathematical models of collective motions of cells under the influence of chemical stimuli. The models are hybrid in the sense that cells are discrete entities given by ODE, while the chemoattractant is considered as a continuous signal which solves a diffusive equation. For this model we prove the mean-field limit in the Wasserstein distance to a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Our approach and results are not based on empirical measures, but rather on marginals of large number of indivi…
Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, in this article we study solutions to the stationary chemotaxis system \[ \begin{cases} 0 = Δn - \nabla\cdot(n\nabla c) \\ 0 = Δc - nc \end{cases} \] in bounded domains $Ω\subset\mathbb{R}^N$, $N\ge 1$, under no-flux boundary conditions…
Well-posedness and uniform-in-time boundedness of classical solutions are investigated for a three-component parabolic system which describes the dynamics of a population of cells interacting with a chemoattractant and a nutrient. The former induces a chemotactic bias in the diffusive motion of the cells and is accounted for by a density-suppressed motility. Well-posedness is first established for generic positive and non-increasing motility functions vanishing at infinity. Growth conditions on the motility function guaranteeing the uniform-in-time boundedness of solutions are next identified.…
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass larg…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0…
This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=Δv- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensional smoothly bounded domains, where the parameters $ξ,χ> 0$, $μ> 0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u > 0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is proved that under the conditions $$m>\frac{2N}{N+{{{\fr…
This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=Δu-χ_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*} v_t=Δv-χ_2\nabla\cdot(v\nabla w), \end{equation*} \begin{equation*} w_t=Δw-(αu+βv)w \end{equation*} in a smooth bounded domain $Ω\subset\mathbb{R}^n$ ($n\geq2$), where the parameters $χ_1$, $χ_2$, $α$ and $β$ are positive. It is proved that if \begin{equation*} \max\{χ_1,χ_2\}\|w(x,0)\|_{L^{\infty}(Ω)}<\sqrt{\frac{2}{n}}π\end{equation*} the problem possesses a uniqu…
We study the chemotaxis model $\partial$ t u = div($\nabla$u -- u$\nabla$w) + $θ$v -- u in (0, $\infty$) x $Ω$, $\partial$ t v = u -- $θ$v in (0, $\infty$) x $Ω$, $\partial$ t w = D$Δ$w -- $α$w + v in (0, $\infty$) x $Ω$, with no-flux boundary conditions in a bounded and smooth domain $Ω$ $\subset$ R 2 , where u and v represent the densities of subpopulations of moving and static individuals of some species, respectively, and w the concentration of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence …
We study an one{dimensional quasilinear system proposed by J. Tello and M. Winkler [19] which models the population dynamics of two competing species attracted by the same chemical. The kinetics terms of the interacting species are chosen to be the Lotka{Volterra type. We prove the existence of global bounded and classical solutions for all chemoattraction rates. Under homogeneous Neumann boundary conditions, we establish the existence of nonconstant steady states by local bifurcation theory. The stability of the bifurcating solutions is also obtained when the diffusivity of both species is la…
We consider a parabolic-elliptic chemotaxis system generalizing \[ \begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{σ-1}\nabla v)\\ & 0 = Δv - v + u \end{split}\end{cases} \] in bounded smooth domains $Ω\subset \mathbb{R}^N$, $N\ge 3$, and with homogeneous Neumann boundary conditions. We show that *) solutions are global and bounded if $σ<m-\frac{N-2}N$ *) solutions are global if $σ\le 0$ *) close to given radially symmetric functions there are many initial data producing unbounded solutions if $σ>m-\frac{N-2}N$. In particular, if $σ\le 0$ and $σ> m-\frac{…
The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= Δu-χ\nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N τv_t=Δv-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We prove that for every $τ>0$, there is $0<χ_τ^*<\frac{b}{2}$ such that for every $0<χ<χ_τ^*$, there exist two positive numbers $2\sqrt a \le c^{*}(χ…
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ τv_t = Δv - v + u \end{cases} \end{align*} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $τ\in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(Ω)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(Ω)$ and the technical condition $m > \frac{n-2 p_0}{n}$ is fulfil…
This paper is a continuation of the paper Berestycki, Wei and Winter \cite{Berestycki2014}. In \cite{Berestycki2014}, the existence of multiple symmetric and asymmetric spike solutions of a chemotaxis system modeling crime pattern formation, suggested by Short, Bertozzi, and Brantingham \cite{Short2010}, has been proved in the one-dimensional case. The problem of stability of these spike solutions has been left open. In this paper, we establish the existence of a single radial symmetric spike solution for the system in the one and two-dimensional cases. The main difficulty is to deal with quas…
This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $Ω\subseteq \mathbb{R}^3$ with smooth bound…
This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)χ(v)\nabla v) +\nabla\cdot(H(u)ξ(w)\nabla w), \quad v_t=d_1Δv+αu-βv, \quad w_t=d_2Δw+γu-δw, \quad x \in Ω,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, α, β, γ, δ>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \…
These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these l…
We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=Δv_1-λ_1v_1+μ_1u,\ x\in\mathbb{R}^N,\\ 0=Δv_2-λ_2v_2+μ_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the steady solution $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ is asymptotica…
Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak--Keller--Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the …
This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - \nabla \cdot (f(u) \nabla v) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-v+g(u…
This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-c+n,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0, \end{array}\right.$$ which describes chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells, where $κ\in \mathbb{R},ϕ\in W^{2,\inf…
This paper investigates the following Keller-Segel-Navier-Stokes system with nonlinear diffusion and rotational flux $$\begin{align}\begin{cases} &n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x, n, c)\nabla c),\quad &x\in Ω, t>0, \\ &c_t+u\cdot\nabla c=Δc-c+n,\quad &x\in Ω, t>0, \\ &u_t+κ(u\cdot\nabla)u+\nabla P=Δu+n\nabla ϕ,\quad &x\in Ω, t>0, \\ &\nabla\cdot u=0,\quad &x\in Ω, t>0, \end{cases}\end{align}$$ where $κ\in \mathbb{R},ϕ\in W^{2,\infty}(Ω)$ and $S$ is a given function with values in $\mathbb{R}^{2\times2}$ which fulfills $$ |S(x,n,c)| \leq C_S $$ with some $C _S > 0$. Systems of this typ…
