On a Chemotaxis-Generalized Navier–Stokes System with Rotational Flux: Global Classical Solutions and Stabilization

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

PurposeThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.Design/methodology/approachThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.FindingsThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.Originality/valueThis article is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Global Classical Solutions and Stabilization in a Two-Dimensional Parabolic-Elliptic Keller–Segel–Stokes System

Global classical solutions and stabilization for a class of competition models with density-dependent motility

As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high‐dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density and the lower bound of the temperature . Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux and the temperature by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: if ; if . This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high‐dimensional space.

Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

This paper concerns with the global classical solution to the Cauchy problem of the nonlinear double dispersive wave equation with strong damping uttuutt + � 2 u cut = d N X i=1 ∂ ∂xi σ i(uxi), where c and d are positive constants. By the contraction mapping principle and priori estimates, we prove that the Cauchy problem admits a unique global classical solution, and by the concavity method, we give the sufficient condi- tions on the blowup of the global solution of the Cauchy problem. Finally, as an application, an example is also given.

Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in [formula omitted

For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

In this paper, we deal with a coupled chemotaxis-fluid model with logistic source \begin{document} $γ n-μ n^2$ \end{document} . We prove the existence of global classical solution for the chemotaxis-Stokes system in a bounded domain \begin{document} $Ω\subset \mathbb R^3$ \end{document} for any large initial data. On the basis of this, we further prove that if \begin{document} $γ>0$ \end{document} , the zero solution is not stable; if \begin{document} $γ = 0$ \end{document} , the zero solution is globally asymptotically stable; and if \begin{document}$ 0 , the nontrivial steady state \begin{document} $\left(\fracγμ, \fracγμ, 0\right)$ \end{document} is globally asymptotically stable.

In this paper, we deal with the Chaplain–Lolas’s model of cancer invasion with tissue remodelling We consider this problem in a bounded domain (N = 2, 3) with zero-flux boundary conditions. We first establish the global existence and uniform boundedness of solutions. Subsequently, we also consider the large time behaviour of solutions, and show that the global classical solution (u, v, w) strongly converges to the semi-trivial steady state in the large time limit if δ > η; and strongly converges to if δ < η. Unfortunately, for the case δ = η, we only prove that (v, w) → (1, 0), and it is hard to obtain the large time limit of u due to lack of uniform boundedness of . It is worth noting that the large time behaviour of solutions for the chemotaxis–haptotaxis model with tissue remodelling has never been touched before, this paper is the first attempt. At last, taking advantage of the large time behaviour of solutions, we also establish the uniform boundedness of solutions in the classical sense.

ABSTRACTWe investigate global existence of classical solution to the three dimensional Vlasov–Poisson system with radiation damping . We prove that a small perturbation of initial datum for a global classical solution verifying certain decay conditions also launches a global solution which has sharp decay. In addition, we prove that a quasineutral initial datum launches a global classical solution.

Global classical solutions in a chemotaxis(-Navier)-Stokes system with indirect signal production

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier–Stokes equations with vacuum in a bounded domain or exterior domain $\Omega$ of $\mathbb{R}^n$($n\ge2$). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1–28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91–129; S.J. Ding, H.Y. Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696–1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension.

Global solutions of advection-dominated accretion flows with two critical points