Quan Wang

We investigate the linear Rayleigh–Taylor instability within the framework of two-dimensional viscous primitive equations (VPE) under hydrostatic approximation, which govern large-scale oceanic and atmospheric flows. For steady quiescent background states with arbitrary density profiles satisfying the Rayleigh–Taylor instability criterion (i.e., heavier fluid overlying lighter fluid), we conduct a rigorous spectral analysis of the linearized perturbation system. By eliminating the vertical velocity and pressure fields, we derive a nonlocal eigenvalue problem governing the horizontal velocity perturbations. To address the intrinsic non-self-adjoint nature of the system, we introduce an auxiliary variational problem with a frozen parameter, which enables us to establish the existence and uniqueness of a positive eigenvalue ${\lambda }_k$ for each admissible horizontal wavenumber k. We further prove that ${\lambda }_k$ is bounded above by a wavenumber-dependent critical value $s_k$, and verify the sharpness of the maximal growth rate defined as $\Lambda :={\sup }_{k\ne 0}{\lambda }_k$. In particular, we show that no small perturbation can grow faster than the exponential rate $e^{\Lambda t}$, and construct explicit normal-mode solutions that achieve growth rates arbitrarily close to $\Lambda$. We also derive complete energy estimates for the linearized system, establishing uniform temporal control of solutions via the bound $e^{2\Lambda t}$. Additionally, we demonstrate that positive viscosity dissipation uniformly bounds all unstable eigenvalues, whereas the zero-dissipation limit gives rise to an unbounded unstable spectrum where eigenvalues grow without bound as the horizontal wavenumber increases. This spectral behavior stands in stark contrast to that of the classical Euler equations, for which all unstable eigenvalues remain uniformly bounded. This fundamental discrepancy reveals that the hydrostatic approximation essentially distorts the intrinsic instability properties of the original fluid system, thereby invalidating its physical validity for numerical simulations of Rayleigh–Taylor instability.

We study the existence and stability of non-trivial steady-state solutions to the two-dimensional incompressible Navier–Stokes equations posed in the annulus $ \Omega = B(0, b) \setminus \overline{B(0, a)} $ with radii $ b>a>0 $. The outer boundary $ \partial B(0, b) $ is subject to the free condition while the inner boundary $ \partial B(0, a) $ obeys a Navier-slip condition with effective slip length $ \alpha > 0 $. Our main results are fourfold. First, we establish global-in-time strong solutions and derive a sharp energy estimate that underpins the subsequent nonlinear instability analysis. Second, for $ \alpha > 0 $, we compute an explicit critical viscosity $ \mu_c: = \mu_c(\alpha, a, b, \mu) $ that separates qualitatively different dynamical behaviours. Third, we precisely characterize the stability properties of the trivial solution in three distinct regimes. The zero solution is globally asymptotically stable in $ H^2 $ if $ \mu > \mu_c $. If $ \mu = \mu_c $, we prove an alternative theorem that completely describes the local dynamics near the trivial state. If $ \mu < \mu_c $, the trivial solution is nonlinearly unstable in every $ L^p $ ($ p \geq 1 $). Finally, we demonstrate that for $ \mu < \mu_c $, the system undergoes a pitchfork bifurcation that generates an infinite family of non-trivial steady states. For generic choices of $ (\alpha, b) $, this bifurcation is supercritical; a measurable subset of parameter space yields subcritical transitions. Notably, all bifurcating solutions share the same topological pattern—a single row of counter-rotating vortices—despite their mathematical non-uniqueness.

This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition \(\mathbf{u}|_{\partial Ω} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -ρ_s \nabla f\). Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh--Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form \(ρ_s = -γf + β\), with \(γ> 0\) and \(β> 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(Ω)\).

We analyze the nonlinear inertial instability of Couette flow under Coriolis forcing in \(\mathbb{R}^{3}\). For the Coriolis coefficient \(f \in (0,1)\), we show that the non-normal operator associated with the linearized system admits only continuous spectrum. Hence, there are no exponentially growing eigenfunctions for the linearized system. Instead, we construct unstable solutions in the form of pseudo-eigenfunctions that exhibit non-ideal spectral properties. Then through a bootstrap argument and resolving the challenges posed by the non-ideal spectral behavior of pseudo-eigenfunctions, we establish the velocity instability of Couette flow in the Hadamard sense for $ f \in \Big(\frac{2}{17} \left(5-2 \sqrt{2}\right), \frac{2}{17} \left(5 + 2 \sqrt{2}\right) \Big)$.

This paper presents a pioneering investigation into the existence of traveling wave solutions for the two-dimensional Euler equations with constant vorticity in a curved annular domain, where gravity acts radially inward. This configuration is highly relevant to astrophysical and equatorial oceanic flows, such as those found in planetary rings and equatorial currents. Unlike traditional water wave models that assume flat beds and vertical gravity, our study more accurately captures the centripetal effects and boundary-driven vorticity inherent in these complex systems. Our main results establish both local and global bifurcation of traveling waves, marking a significant advancement in the field. First, through a local bifurcation analysis near a trivial solution, we identify a critical parameter \(α_c\) and prove the existence of a smooth branch of small-amplitude solutions. The bifurcation is shown to be pitchfork-type, with its direction (subcritical or supercritical) determined by the sign of an explicit parameter \(\mathcal{O}\). This finding provides a nuanced understanding of the wave behavior under varying conditions. Second, we obtain a global bifurcation result using a modified Leray-Schauder degree theory. This result demonstrates that the local branch extends to large-amplitude waves. This comprehensive analysis offers a holistic view of the traveling wave solutions in this complex domain. Finally, numerical examples illustrate the theoretical bifurcation types, confirming both supercritical and subcritical regimes. These examples highlight the practical applicability of our results.