Rotation is a crucial characteristic of fluid flow in the atmosphere and oceans, which is present in nearly all meteorological and geophysical models. The global existence of solutions to the 3D Navier-Stokes equations with large rotation has been established through the dispersion effect resulting from Coriolis force (i.e., rotation). In this paper, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number Re . Our aim is to determine the stability threshold index on Re : the maximum range of perturbations within which the solution remains stable. Initially, we examine the linear stability properties of the perturbed system. By comparing our findings with the classical 3D Navier-Stokes equations, we note that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow. Additionally, we discover that the rotation effect serves as a restoring mechanism that induces dispersion for inertial waves, effectively counteracting lift-up effects observed at zero frequency velocity in the Bradshaw-Richardson stable regime, and this dispersion mechanism exhibits favorable algebraic decay properties distinct from those seen in the classical 3D Navier-Stokes equations, which indicate that rotation strengthens the stability effect of fluid dynamics. Consequently, we demonstrate that if the Sobolev norm of the initial data in H^σ with σ >9/2 is controlled by 𝒪(Re^-1) , then the solution to the 3D Navier-Stokes equations with rotation is global in time without transitioning away from Couette flow.

Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers (𝐑𝐞) in the periodical domain 𝕋×ℝ×𝕋, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of u^2 and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation u_in_H^σ < δ𝐑𝐞^-2 for any σ > 9/2 and some δ=δ(σ)>0 depending only on σ.

This paper is concerned with an initial and boundary value problem for planar compressible magnetohydro dynamics with temperature-dependent transport coefficients. In the case when the viscosity mu(theta) = theta(alpha), the magnetic diffusivity v(theta) = theta(alpha) and the heat-conductivity k(theta) = theta(beta) with alpha, beta is an element of [0, infinity), we prove the global existence of strong solution under some restrictions on the growth exponent a and the initial norms. As a byproduct, the exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if alpha >= 0 is small, and the growth exponent of heat-conductivity beta >= 0 can be arbitrarily large.

This paper is concerned with the Hall effect on the asymptotic stability of the global solutions to an initial-boundary value problem of the planar compressible MHD system. In the case when the heat conductivity depends on the temperature in the form κ (θ )=θ ^β with β∈ (0,+∞ ) , we show that the global large solutions decay exponentially in time to the equilibrium states without any restriction on the Hall coefficient ε . The exponential stability of the global large solutions still holds when the heat conductivity is a positive constant, provided the Hall coefficient is suitably small. As by-products, the vanishing limit of Hall coefficient is also justified in both cases.

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 solution is nonlinearly exponentially stable as time tends to infinity. This is the first result obtaining the exponential stability behavior of strong solutions to the equations of a planar magnetohydrodynamic compressible flow without any smallness conditions on the data. Our result can be regarded as a natural generalization of the previous ones for the compressible Navier-Stokes system to MHD system with either constant heat-conductivity or nonlinear and temperature-depending heat-conductivity. As a direct consequence, it is shown that the global strong solution to the constant heat-conductivity MHD system whose existence is obtained by Kazhikhov in 1987 is nonlinearly exponentially stable.

We study the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature. In particular, we obtain the global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. This result generalizes the classical one of the compressible Navier–Stokes system with constant viscosity and heat conductivity [Kazhikhov, Siberian. Math. J. 23, 44–49 (1982)] to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic and hydrodynamic effects is complex and the motion of the flow has large oscillations.

This paper is concerned with an initial and boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients. In the case when the viscosity μ(θ)=θα and the heat-conductivity κ(θ)=θβ with α,β∈[0,∞), we prove the global-in-time existence of strong solutions under some assumptions on the growth exponent α and the initial data. As a byproduct, the nonlinearly exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if α≥0 is small, and the growth exponent β≥0 can be arbitrarily large.

We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenerate and nonlinear one under stress-free and thermally insulated boundary conditions.

For the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature, we obtain global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. Our result generalizes the classical one of the compressible Navier-Stokes system with constant viscosity and heat conductivity ([Kazhikhov. Siberian Math. J. (1982)]) to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic effects and hydrodynamic is complex and the motion of the flow has large oscillations.

We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case (Kazhikhov 1987 Boundary Value Problems for Equations of Mathematical Physics (Krasnoyarsk)), we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the hydrodynamic and magnetodynamic effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov’s theory for the constant viscosity and heat conductivity case to that of nonlinear viscosity and degenerate heat-conductivity.