3D Micropolar Equations Research Articles

The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term

<abstract><p>This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the $ ((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is demonstrated. Meanwhile, the $ (V_{1}\times V_{2}, \mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is proved. Finally, the existence of a $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and a $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and the $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are verified to be the same.</p></abstract>

Well-posedness and large time decay for the 3D micropolar equations with only velocity dissipation

This paper studies the well-posedness and large time behavior of solutions to the 3D micropolar equations with only velocity dissipation. The global weak solutions and regular small solution are examined by exploring the higher order regularity properties of two related linearized problems. Due to some new observation on the special structure of equations in an ingenious way, we introduce an alternative analysis argument and give a detailed description on the large time decay rates for the weak and regular small solutions.

Trajectory Attractors of the 3D Micropolar Equations with Damping Term

Trajectory Attractors of the 3D Micropolar Equations with Damping Term

Large time behavior of solutions to the 3D micropolar equations with nonlinear damping

This paper is concerned with 3D micropolar equations with a velocity damping term σ|u|β−1u. We mainly focus on the large time decay of the L2-norm of weak solutions with β>145 and consider the upper bounds of the derivatives of the strong solution to the system with β≥3. Moreover, the asymptotic stability for the large solutions of (1.1) and the error estimate between (1.1) and the classic micropolar equations is also investigated.

A note on global existence of strong solution to the 3D micropolar equations with a damping term

This paper studies the Cauchy problem of the 3D incompressible micropolar equations with a damping term sigma |u|^{beta -1}u (sigma >0, 1le beta <3). It is shown that the strong solutions exist globally for any 1le beta <3.

Global attractors of the 3D micropolar equations with damping term

The 3D micropolar system with a damping term is considered by the uniform estimates. In this paper, global attractors of the 3D micropolar equations with damping term are proved for \begin{document}$ 3 with any \begin{document}$ \sigma>0 $\end{document} .

Global well-posedness of the 3D micropolar equations with partial viscosity and damping

In this paper, we are concerned with the Cauchy problem of the 3D incompressible micropolar equations with partial viscosity and damping. We establish the global existence and uniqueness of the solution with a general initial data.

Global Regularity of the Three-Dimensional Fractional Micropolar Equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( \Delta)^{\alpha}u$ and $(-\Delta)^{\beta}w$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $\alpha\geq \frac{5}{4}$, $\beta\geq 0$ and $\alpha+\beta\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

Asymptotic stability of planar rarefaction wave to 3D micropolar equations

We are concerned with the large-time behavior of the Cauchy problem to the 3d micropolar fluids in an infinite long flat nozzle domain R×T2. In one dimensional case, this system tends time-asymptotically to the Navier–Stokes equations. That is to say, the basic wave patterns to the compressible micropolar fluids model are stable. Hence, in this paper we consider the nonlinear stability of planar rarefaction wave to the corresponding three dimensional model. Some cancellations on the flux terms and viscous terms are crucial. Moreover, a proper combining of damping term and rotation terms can provide an extra regularity of w.

Global regularity for the 3D micropolar equations

In this paper a global weak solution of the 3D generalized micropolar equations with any α≥1 and β>0 corresponding to any L2 initial data are considered. Actually, weak solutions associated with α≥54 and β≥1 are global classical solutions when their initial data are sufficiently smooth.

Global existence of strong solution to the 3D micropolar equations with a damping term

In this note we establish the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional incompressible micropolar equations with a nonlinear damping term.

Global regularity for d-dimensional micropolar equations with fractional dissipation

ABSTRACTThis paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.

An effective model for the lubrication with micropolar fluid

In this paper, using asymptotic analysis, we study the lubrication process with incompressible micropolar fluid. Starting from 3D micropolar equations, we derive the higher-order asymptotic model explicitly acknowledging the microstructure effects. The effective equations are similar to the Brinkman model for porous medium flow.