Ibrahim, S Lin, Q Titi, ES

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finit...

Li, Ji Liu, Yue
Published in
Annals of PDE

We study the stability of smooth and peaked solitary waves to the modified Camassa-Holm equation. This quasilinear equation with cubic nonlinearity is completely integrable and arises as a model for the unidirectional propagation of shallow water waves. Based on the phase portrait analysis, we demonstrate the existence of unique localized smooth so...

Wang, Yong Wu, Wenpei
Published in
Advances in Nonlinear Analysis

We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H2 space is obtained. Moreover, we prove that the solution decays to the equilib...

Ke, Xueli Yuan, Baoquan Xiao, Yaomin
Published in
Acta Mathematica Scientia

This paper is concerned with a stability problem on perturbations near a physically important steady state solution of the 3D MHD system. We obtain three major results. The first assesses the existence of global solutions with small initial data. Second, we derive the temporal decay estimate of the solution in the L2-norm, where to prove the result...

Gao, Junpei Cui, Haibo
Published in
Acta Mathematica Scientia

We investigate the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half line ℝ+ ≔ (0, ∞). Inspired by the relationship between a micropolar fluid model and Navier-Stokes equations, we prove that the composite wave consisting of the transonic boundary layer solution, the 1-rarefaction wave, ...

Blauth, Sebastian Leithäuser, Christian Pinnau, René
Published in
Journal of Engineering Mathematics

We consider the optimization of a chemical microchannel reactor by means of PDE-constrained optimization techniques, using the example of the Sabatier reaction. To model the chemically reacting flow in the microchannels, we introduce a three- and a one-dimensional model. As these are given by strongly coupled and highly nonlinear systems of partial...

Figueiredo, G. M. Ruviaro, R. Moura, E. L. de Oliveira Junior, J. C.
Published in
Mediterranean Journal of Mathematics

We are concerned on the quasilinear problems where Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^{N}$$\end{document} is a bounded domai...

Dong, Boqing Wu, Jiahong Xu, Xiaojing Zhu, Ning
Published in
Calculus of Variations and Partial Differential Equations

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal th...

Huang, Bin Shi, Xiaoding Sun, Ying
Published in
Journal of Mathematical Fluid Mechanics

For the strong solutions to the equations of a planar magnetohydrodynamic compressible flow with the heat conductivity proportional to a nonnegative power of the temperature, we first prove that both the specific volume and the temperature are proved to be bounded from below and above independently of time. Then, we also show that the global strong...

Zillinger, Christian
Published in
Journal of Nonlinear Science

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \use...