Vanishing viscosity limit of the 3D incompressible micropolar equations in a bounded domain

In this paper, we study the vanishing viscosity limit for the 3D incompressible micropolar equations in a flat domain with boundary conditions. We prove the existence of the global weak solution of the micropolar equations and obtain the uniform estimate of the strong solution. Furthermore, we establish the convergence rate from the solution of the micropolar equations to that of the ideal micropolar equations as all viscosities tend to zero (i.e., (ε,χ,γ,κ) → 0).

Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form "Equation missing"where $$\kappa \ge 0$$ κ ≥ 0 , $$\mu >0$$ μ > 0 and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N with $$N\in \{2,3\}$$ N ∈ { 2 , 3 } is a prescribed time-independent nonnegative function $$c_*\in C^{2}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) $$ c ∗ ∈ C 2 Ω ¯ . Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.

Qualitative analysis of solutions for a parabolic type Kirchhoff equation with logarithmic nonlinearity

In this work, we investigate the initial boundary-value problem for a parabolic type Kirchhoff equation with logarithmic nonlinearity. We get the existence of global weak solution, by the potential wells method and energy method. Also, we get results of the decay and finite time blow up of the weak solutions.

On the 2D Ericksen–Leslie equations with anisotropic energy and external forces

In this paper we consider the 2D Ericksen–Leslie equations which describe the hydrodynamics of nematic liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is regular as long as the data are regular.

Global weak solution to a generic reaction-diffusion nonlinear parabolic system

We consider a new generic reaction-diffusion system, given as the following form: ∂u/∂t - div(g(│(∇u_σ)│)∇u)=f(t,x,u,v,∇v), in Q_T ∂v/∂t - d_v Δv=p(t,x,u,v,∇u), in Q_T u(0,.)=u_0, v(0,.)=v_0, in Ω (1) ∂u/∂η=0, ∂v/∂η=0, in ∑_T. Where Ω=]0,1[?×]0,1[, Q_T =]0,T [? and T =]0,T [?, (T > 0), η is an outward normal to domain Ω and u_0, v_0 is the image to be processed, x ∈Ω, σ >0, ∇u_σ= u∗ ∇G_σ and G_σ= 1/√2πσ exp(-│x│^2/4σ). In this study we are going to proof that there is a global weak solution to the ptoblem (1), we truncate the system and show that it can be solved by using Schauder fixed point theorem in Banach spaces. Finally by making some estimations, we prove that the solution of the truncated system converge to the solution of the problem.

Analysis of long-term solution of chemotactic model with indirect signal consumption in three-dimensional case

In this paper, we consider the chemotaxis model u_t&=\Delta u-\nabla\cdot(u\nabla v),& \qquad x\in\Omega,\,t>0,v_t&=\Delta v-vw,& \qquad x\in\Omega,\,t>0,w_t&=-\delta w+u,& \qquad x\in\Omega,\,t>0,under homogeneous Neumann boundary conditions in a bounded and convex domain $\Om\subset \mathbb{R}^3$ with smooth boundary, where $\delta>0$ is a given parameter. It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists $T>0$ such that the solution $(u,v,w)$ is bounded and smooth in $\Om\times(T,\infty)$. Furthermore, it is asserted that such solutions approach spatially constant equilibria in the large time limit.

Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian

<p style='text-indent:20px;'>We consider a two-species chemotaxis-Navier-Stokes system with <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian in three-dimensional smooth bounded domains. It is proved that for any <inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>, the problem admits a global weak solution.</p>