We are concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system with degenerate heat-conductivity describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. Both the specific volume and temperature are proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.

In this paper, we study the convergence of the solutions to the Dirichlet problem of the incompressible micropolar fluid equations with full anisotropic dissipation toward the solution to the ideal micropolar fluid equations in the upper half-plane. By choosing suitable correctors, we find that if the vertical dissipation of horizontal fluid velocity, the vertical angular viscosity and the micro-rotation viscosity vanish more quickly than the others, the vanishing dissipation limit exists in L^∞([0,T];L^2(ℝ^2_+)) . In addition, we deal with the difficulty caused by the micro-rotation viscosity. Further, we obtain the convergence rate.

We investigate the stability and large-time behavior of solutions to the three-dimensional incompressible Navier–Stokes equations with horizontal fractional dissipation of order α∈ (0,1] near a constant background state v^(0) = A . For sufficiently small initial perturbations in H^3(ℝ^3) , we establish global existence and uniform bounds for the solutions. Moreover, for a suitable range of parameters α and σ , we prove that these solutions exhibit optimal time-decay rates, with the third component decaying faster than the horizontal components, reflecting an enhanced dissipation phenomenon. The proofs rely on a combination of anisotropic inequalities, energy estimates, and integral representations associated with the fractional horizontal heat semigroup. These results extend classical well-posedness and decay theory for the Navier–Stokes equations to the setting of partial dissipation and provide a framework for studying related geophysical and anisotropic fluid models.

We investigate a fully dissipative isentropic compressible micropolar fluid in a finite-depth and horizontally periodic domain of three dimensional, with a free-moving boundary and a fixed solid boundary. The fluid is governed by gravity-driven isentropic compressible micropolar equations, and the surface tension is ignored on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem of the compressible micropolar equations. Additionally, we prove that the solution decays to a nontrivial equilibrium algebraically, provided that the initial data are sufficiently close to the equilibrium state. (c) 2025 Published by Elsevier Inc.

The Tropical Climate Model (TCM) is a simplified system that captures key aspects of equatorial atmospheric dynamics through the interaction of barotropic and baroclinic velocity modes with temperature fields. This study focuses on the nonlinear stability of Couette flow in a two-dimensional TCM with only partial dissipation. Two main difficulties arise: the absence of full dissipation, and the lack of a divergencefree condition for the baroclinic velocity. To address these challenges, we develop a refined Fourier multiplier approach that captures enhanced dissipation via the interaction between the shear-induced mixing term and vertical viscosity. Furthermore, this paper introduces new techniques to handle terms involving non-divergence-free components and exploits key couplings within the system to control potentially unstable linear terms. Under appropriate smallness conditions on the initial perturbations in anisotropic Sobolev spaces, we rigorously establish the nonlinear stability of the Couette flow and identify a possible precise transition threshold for stability. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.

In this paper, we consider the 3D Boussinesq boundary layer system in R+ x R2, which is a coupling of the Prandtl type equations and a thermal layer equation due to the coupling of velocity and temperature in Boussinesq equations. We observe that there is also a cancellation mechanism in the temperature equation, which has been applied to the Prandtl equations in Li et al. (2022) [14]. Utilizing these cancellation mechanisms and constructing good unknowns, we overcome the loss of derivative arising in not only the velocity equations but also the temperature equation, then we show the local well-posedness of the Boussinesq boundary layer system in Gevrey function spaces. Furthermore, we obtain the optimal Gevrey index 2. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper establishes the global existence of non-negative weak solutions to a two-dimensional, fourth-order nonlinear degenerate parabolic equation modeling surface-tension-driven convection in thin fluid films. First, we construct a regularized approximate problem and prove its solvability via the Galerkin method. Utilizing energy and entropy functionals alongside a singular entropy condition 1/h_0 ∈ L^1(Ω), we secure uniform a priori bounds for higher-order spatial and time derivatives. These bounds enable the use of the Aubin-Lions lemma and Gagliardo-Nirenberg inequalities to achieve strong compactness and essential L^6-integrability. Furthermore, we adopt the Alber-Zhu framework to rigorously define higher-order local weak derivatives and pass to the limit. Finally, we prove the limit function is non-negative, confirming it as a global weak solution to the original problem.

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.

This paper examines the stability threshold at high Reynolds numbers Re for the three-dimensional Boussinesq equations with rotation on the domain Ω={(x, y, z)∈𝕋×ℝ×𝕋} around the Couette flow (y,0,0) and the vertically stratified temperature Θ_s=1+α^2 z. For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy u_in_H^N+2∩ W^N+3,1+θ_in_H^N+1∩ W^N+3,1<δ𝐑𝐞^-14/15, for any N≥ 11 and some δ>0 independent of 𝐑𝐞, then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by (-Δ)^α/2u and (-Δ)^β/2 θ. Attention is focused on the subcritical regime where α+ β>1. The case α>2/3 was recently settled in a joint work of the authors [Math. Ann., 391 (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case α≤2/3. We obtain the sharpest regularity result by minimizing assumptions on α and β. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

This paper investigates the global existence and non-negativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection phenomena in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and a corresponding Galerkin approximating scheme. We first establish the existence of solutions to the approximate problem. Subsequently, by constructing specialized energy and entropy functionals, we derive uniform a priori estimates for the approximating solutions. Leveraging the Aubin-Lions compactness lemma, we pass to the limit and establish the non-negativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.

We study the hydrostatic approximation for the three-dimensional Boussinesq equations of damped wave type. This is a mixed degenerate system coupled with parabolic and hyperbolic equations. Compared with the purely hyperbolic hydrostatic Navier--Stokes equations, the parabolic equation for temperature will lead to an extra loss of derivatives. In the setting of Gevrey space with index 7/4, we prove the local well-posedness and the corresponding hydrostatic limit for the three-dimensional Boussinesq equations of damped wave type.

This paper studies the stability and large-time behavior of perturbations around a large, constant magnetic field in a periodic, infinite channel under specific symmetry constraints. Mathematically, the perturbations are governed by the 2D incompressible magnetohydrodynamic equations with no velocity dissipation and only horizontal magnetic diffusion. This stability result is sharp in the sense that removing this horizontal magnetic diffusion leads to instability. The proof is nontrivial and involves delicate construction of a time-weighted energy functional. Our result rigorously confirms the stabilizing effect of a background magnetic field on electrically conducting fluids.

This paper presents some of the sharpest global existence and regularity results on the two-dimensional incompressible Boussinesq equations with fractional dissipation, Lambda(alpha)u and Lambda(beta)theta, where Lambda=root-Delta is the Zygmund operator. For the subcritical regime alpha+beta>1 with alpha > 2/3, any initial data in the Sobolev space H-s(R-2) with s > 2 leads to a unique global solution. For any (alpha, beta)in the critical regime alpha+beta=1 with alpha > 2/3, an extra smallness condition on the L-infinity-norm of the initial temperature wouldalso guarantee the global regularity. This paper introduces an iterative procedure tominimize the dissipation requirement.

The magnetohydrodynamic (MHD) equations admit a physically relevant steady-state solution given by Couette flow combined with a background magnetic field. This paper investigates the nonlinear stability of perturbations around this steady state in an MHD system with partial dissipation. We establish the desired stability when the initial perturbation, measured in terms of the vorticity omega(0) and the current density j0, satisfies divided by divided by(omega(0),j(0))divided by(H)b <= epsilon nu(1/3), where epsilon>0 is a suitable constant, b >= 2 and nu denotes the viscosity. The exponent 1/3 appears to be optimal and indicates the sharp stability threshold for this MHD system. The proof analyzes the perturbation dynamics over two distinct time intervals and employs the Fourier multiplier method without a change of variables.

In atmospheric and oceanic fluid models, as exemplified by the Boussinesq system, temperature plays a crucial role. This paper investigates the limit of small thermal diffusivity for the initial-boundary value problem of the 2D Boussinesq equations with fixed viscosity, featuring non-slip velocity boundary conditions and prescribed temperature data. We focus on the boundary layer phenomena associated with the Boussinesq equations. A key observation is that when the boundary temperature varies with time, a dominant thermal boundary layer emerges, coexisting with a weaker velocity boundary layer in the small diffusivity regime. Additionally, using energy methods, we rigorously justify the solution expansions of the Boussinesq system in the LC-norm. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper intends to understand the long-time existence and stability of solutions to an Euler-like equation. An Euler-like equation is the 2D incompressible Euler equation with an extra singular integral operator (SIO) type term. In contrast to the 2D Euler equation, the vorticity to the 2D Euler-like equation is not known to be bounded due to the unboundedness of the SIO on the space L infinity. As a consequence, classical Yudovich theory fails on the Euler-like equation. The global existence, regularity and stability problems on the Euler-like equation are generally open. This paper makes progress on an Euler-like equation arising in the study of several fluids. We establish a long-time existence and stability result. When the Sobolev size of the initial data is of order epsilon, the solution is shown to live on a time interval of the size 1/epsilon 2. When the initial data is restricted to a class with special symmetry, we obtain the global existence and nonlinear stability. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper concerns the Cauchy problem to the compressible magnetohydrodynamic equations in R-2 with the constant state of density at far field being vacuum or nonvacuum. Under the conditions that the adiabatic constant gamma > 1, the shear viscosity coefficient mu is a positive constant, and the bulk one lambda(rho) = rho(beta) with beta > 4/3, we establish the global existence and uniqueness of strong solutions. In particular, the initial data can be arbitrarily large and the density is allowed to vanish initially. These results generalize and improve previous ones by Huang-Li (2022) and Jiu-Wang-Xin (2018) for compressible Navier-Stokes equations. This paper introduces some key weighted estimates on H and presents some delicate analysis to exploit the decay properties of solutions due to the strong coupling and interplay interaction. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The goal of this work is to study the Boussinesq equations for an incompressible fluid in R2, with diffusion modeled by fractional Laplacian. The existence, the uniqueness and the regularity of solution has been proved.