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Changzhen Sun

Publications and source records attributed to Changzhen Sun.

14 recordsLinked to original sources
arXiv2025

Transition threshold for the Navier-Stokes-Coriolis system at high Reynolds numbers

The transition mechanism from laminar flow to turbulent flow is a central problem in hydrodynamic stability theory. To shed light on this transition mechanism, Trefethen et al.({\it \small Science 1993}) proposed the transition threshold problem, aiming to quantify the magnitude of perturbations required to trigger instability and determine their scaling with the Reynolds number. In this paper, we investigate the transition threshold of Couette flow for the three-dimensional incompressible Navier-Stokes-Coriolis system in the high Reynolds number regime ($\mathrm{Re}\gg 1$). By exploiting the combined effects of rotation (dispersion) and mixing mechanisms, we derive an improved stability threshold scaling in $\mathrm{Re}$. Precisely, we show that if the initial perturbation satisfies $$\|v_{in}-(y, 0, 0)\|_{\tilde{H}(\mathbb T \times \mathbb D)}\leq \epsilon_0 \,\mathrm{Re}^{-\alpha},$$ with any $\alpha>\frac 23$ and $\tilde{H}=H^6 \cap W^{3,1}$ for $\mathbb D=\mathbb{R}^2$, and with any $\alpha \geq\frac 56$ and $\tilde{H}=H^6$ for $\mathbb D=\mathbb{R}\times\mathbb{T}$, the corresponding solution of the Navier-Stokes-Coriolis system exists globally in time and remains asymptotically close to the Couette flow. The main analytical challenge arises from the anisotropic nature of the estimates for the zero modes and from the interactions between zero and non-zero modes, which we address using an anisotropic Sobolev space directly tailored to the zero modes. Additionally, we introduce a new dispersive structure for the zero modes and derive suitable Strichartz-type estimates. These tools enable us to exploit both the nonlinear structure and the improved dispersive behavior of certain good components of the zero modes, which play a crucial role in achieving the improved stability threshold.

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arXiv2025

On the spectral stability of periodic capillary-gravity waves

In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral stability of the waves. This result aligns with earlier formal analyses by Djordjevi\'c \& Redekopp [15] and Ablowitz \& Segur [1], which employed the nonlinear Schr\"odinger approximation in the modulational regime. In particular, we show that instability is excluded near spectral crossings away from the origin when the surface tension is positive and the inverse square of the Froude number $\alpha\in(0,1),$ which results from the fact that the corresponding Krein signatures are identical. It is also shown that there exists $\alpha_1 = (23 - 3\sqrt{41})/8$ and a curve $\beta: (\alpha_1, 1]\rightarrow \mathbb{R}_{+},$ such that for any $\alpha \in (\alpha_1, 1]$, small amplitude periodic waves are spectrally stable when $\beta > \beta(\alpha)$. These findings highlight the stabilizing effect of surface tension on periodic capillary-gravity waves.

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arXiv2025

Transverse asymptotic stability of line solitary waves for the Ionic Euler-Poisson system

We prove the linear and nonlinear asymptotic stability of small amplitude one-dimensional solitary waves submitted to small localized irrotational perturbations in the three dimensional Euler-Poisson system describing the dynamics of ions. In particular, in this regime, we obtain the existence of global smooth solutions and describe their asymptotic behavior.

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arXiv2025

Incompressible and vanishing vertical viscosity limit for the compressible Navier-Stokes system with Dirichlet boundary conditions

In this paper, we show the incompressible and vanishing vertical viscosity limits for the strong solutions to the isentropic compressible Navier-Stokes system with anistropic dissipation, in a domain with Dirichlet boundary conditions in the general setting of ill-prepared initial data. We establish the uniform regularity estimates with respect to the Mach number $\epsilon$ and the vertical viscosity $\nu$ so that the solution exists on a uniform time interval $[0,T_0]$ independent of these parameters. The key steps toward this goal are the careful construction of the approximate solution in the presence of both fast oscillations and two kinds of boundary layers together with the stability analysis of the remainder. In the process, it is also shown that the solutions of the compressible systems converge to those of the incompressible system with only horizontal dissipation, after removing the fast waves whose horizontal derivative is bounded in $L_{T_0}^2L_x^2$ by $\min\{1, (\epsilon/\nu)^{\frac14}\}.$

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arXiv2024

Transverse linear stability of one-dimensional solitary gravity water waves

In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates. To achieve this, we use different arguments depending on the size of the transverse frequencies. For high transverse frequencies, we use reductions based on pseudodifferential calculus, for intermediate ones, we use an energy-based approach relying on the design of various appropriate energy functionals for different regimes of longitudinal frequencies and for low frequencies, we use the KP-II approximation. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.

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arXiv2023

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations

In this note, we extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R} to small-amplitude periodic traveling waves of the generalized Korteweg-de Vries equations that are not subject to Benjamin--Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability.

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arXiv2022

Uniform regularity in the low Mach number and inviscid limits for the full Navier-Stokes system in domains with boundaries

In the present work, motivated by the studies on the low Mach number limit problem, we establish uniform regularity estimates with respect to the Mach number for the non-isentropic compressible Navier-Stokes system in smooth domains with Navier-slip boundary conditions, in the general case of ill-prepared initial data. The thermal conduction is taken into account and the large variation of temperature is allowed. Moreover, the obtained regularity estimates are also uniform in the Reynolds number $\text{Re}\in[1,+\infty),$ Péclet number $\text{Pe}\in [1,+\infty),$ provided $$\big|\frac{1}{\text{Re}}-\frac{ι_0}{\text{Pe}}\big|\lesssim \frac{1}{\text{Pe}^{\frac{1}{2}}}\frac{1}{\text{Re}},$$ where $ι_0$ is a fixed constant independent of the Mach number, Reynolds number, and Péclet number. The large temperature variation as well as the interactions of two kinds of boundary layers are the main obstacles to the proof.

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arXiv2022

Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson system

The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis and, beyond specific computations, newly devised arguments. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs.

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arXiv2021

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $\sqrt{\varepsilon}$, $\varepsilon$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

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arXiv2021

Uniform regularity for the compressible Navier-Stokes system with low Mach number in bounded domains

We establish uniform with respect to the Mach number regularity estimates for the isentropic compressible Navier-Stokes system in smooth domains with Navier-slip condition on the boundary in the general case of ill-prepared initial data. To match the boundary layer effects due to the fast oscillations and the ill-prepared initial data assumption, we prove uniform estimates in an anisotropic functional framework with only one normal derivative close to the boundary. This allows to prove the local existence of a strong solution on a time interval independent of the Mach number and to justify the incompressible limit through a simple compactness argument.

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arXiv2020

Long-term regularity of two dimensional Navier-Stokes-Poisson equations

This manuscript is devoted to the long-term regularity of the 2-d Navier-Stokes-Poisson system. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the rotational part of the initial velocity is assumed to be small compared to $\varepsilon$. We then show that the lifespan of the system $T^{\varepsilon}$ satisfies $T^{\varepsilon}>\varepsilon^{-(1-\vartheta)}$, where the small constant $\vartheta$ is the size of the initial perturbation in some suitable space. The normal form transformation and the classical parabolic energy estimates are the main ingredients of the proof.

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arXiv2020

Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

We prove a stability result of constant equilibria for the three-dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the incompressible part of the initial velocity is assumed to be small compared to $\varepsilon$. We then get a unique global smooth solution. We also prove a uniform in $\varepsilon$ time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at $\varepsilon$ fixed and the dispersive techniques (dispersive estimates and normal form transformation) that are useful for the inviscid irrotational system.

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arXiv2020

Large time existence of Euler-Korteweg equations and two-fluid Euler-Maxwell equations with vorticity

The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg system and the two-fluid Euler Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5/2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with small irrotational initial data.

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