arXiv subjects
Publications and source records attributed to Weiren Zhao.
We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in $\mathbb{T}\times [0,1]$ when the initial perturbation is in Gevrey-$\frac{1}{s}$ ($\frac{1}{2}<s<1$) class with compact support.
In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s<2$. More precisely, we study the Rayleigh operator $\mathcal{L}_{U_{m,γ}}= U_{m,γ}\partial_x-U''_{m,γ}\partial_xΔ^{-1}$ associated with perturbed shear flow $(U_{m,γ}(y),0)$ in a finite channel $\mathbb{T}_{2π}\times [-1,1]$ where $U_{m,γ}(y)=U(y)+mγ^2\widetildeΓ(y/γ)$ with $U(y)$ being a stable monotonic shear flow and $\big\{mγ^2\widetildeΓ(y/γ)\big\}_{m\geq 0}$ being a family of perturbations parameterized by $m$. We prove that there exists $m_*$ such that for $0\leq m m_*$ which also leads to instability.
In this paper, we study the long-time behavior of the solutions to the two-dimensional incompressible free Navier Stokes equation (without forcing) with small viscosity $ν$, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold $ν^{\frac{1}{2}}$ for perturbations in the critical space $H^{log}_xL^2_y$. Specifically, if the initial velocity $V_{in}$ and the corresponding vorticity $W_{in}$ are $ν^{\frac{1}{2}}$-close to the shear flow $(b_{in}(y),0)$ in the critical space, i.e., $\|V_{in}-(b_{in}(y),0)\|_{L_{x,y}^2}+\|W_{in}-(-\partial_yb_{in})\|_{H^{log}_xL^2_y}\leq εν^{\frac{1}{2}}$, then the velocity $V(t)$ stay $ν^{\frac{1}{2}}$-close to a shear flow $(b(t,y),0)$ that solves the free heat equation $(\partial_t-ν\partial_{yy})b(t,y)=0$. We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense $\|W_{\neq}\|_{L^2}\lesssim εν^{\frac{1}{2}}e^{-cν^{\frac{1}{3}}t}$ and $\|V_{\neq}\|_{L^2_tL^2_{x,y}}\lesssim εν^{\frac{1}{2}}$. In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator $b(t,y)\partial_x-\partial_{yy}b(t,y)\partial_xΔ^{-1}$, which could be useful in future studies.
In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $ν>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.
In this paper, we prove the asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system, when the perturbations are in Gevrey-$\frac{1}{s}$, $(\frac12<s\leq 1)$ and of size smaller than the resistivity coefficient $μ$. More precisely, we prove (1) the $μ^{-\frac13}$-amplification of the perturbed vorticity, namely, the size of the vorticity grows from $\|ω_{\mathrm{in}}\|_{\mathcal{G}^{λ_{0}}}\lesssim μ$ to $\|ω_{\infty}\|_{\mathcal{G}^{λ'}}\lesssim μ^{\frac23}$; (2) the polynomial decay of the perturbed current density, namely, $\left\|j_{\neq}\right\|_{L^2}\lesssim \frac{c_0 }{\langle t\rangle^2 }\min\left\{μ^{-\frac13},\langle t \rangle\right\}$; (3) and the damping for the perturbed velocity and magnetic field, namely, \[ \left\|(u^1_{\neq},b^1_{\neq})\right\|_{L^2}\lesssim \frac{c_0μ}{\langle t\rangle }\min\left\{μ^{-\frac13},\langle t \rangle\right\}, \quad \left\|(u^2,b^2)\right\|_{L^2}\lesssim \frac{c_0μ}{\langle t\rangle^2 }\min\left\{μ^{-\frac13},\langle t \rangle\right\}. \] We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.
In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity $ν>0$ in $\mathbb{T}\times\mathbb{R}$. It's generally known the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the size and the regularity of the initial perturbation that makes the nonlinear asymptotic stability holds. More precisely, we proved that if the initial perturbation is in some Gevrey-$\frac{1}{s}$ class with size $εν^β$ where $s\geq \frac{1-3β}{2-3β}$ and $β\in [0,\frac{1}{3}]$, then the nonlinear asymptotic stability holds.
In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely the Richardson number $γ^2>\frac{1}{4}$. Precisely, we prove that if the initial perturbation $(u_{in},\vartheta_{in})$ of the Couette flow $v_s=(y,0)$ and the linear temperature $ρ_s=-γ^2y+1$ satisfies $\|u_{in}\|_{H^{s+1}}+\|\vartheta_{in}\|_{H^{s+2}}\leq ε_0ν^{\frac{1}{2}}$, then the asymptotic stability holds.
In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $ν$ and small thermal diffusion $μ$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},ρ_{in})$ satisfies $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{ν,μ\}^{\f12}$ and $\|ρ_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{ν,μ\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $ν, μ$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min\{ν,μ\}^{\f12})$ of the Couette flow, and approaches to Couette flow as $t\to\infty$; the temperature remains within $O(\min\{ν,μ\}^{\f{11}{12}})$ of the constant $1$, and approaches to $1$ as $t\to\infty$.
In this paper, we study the metastability for the 2-D linearized dissipative quasi-geostrophic equation with small viscosity $ν$ around the quasi steady state $θ_{sin}=e^{-νt}\sin y$. We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.
In this paper, we study the long-time behavior of the solution for the linearized ideal MHD around sheared velocity and magnetic field under Stern stability condition. We prove that the velocity and magnetic field will converge to sheared velocity and magnetic field as time approaches infinity. Moreover a new depletion phenomenon is proved: the horizontal velocity and magnetic field at the critical points will decay to 0 as time approaches infinity.
In this paper, we prove the stability of Couette flow for 2D Navier-Stokes Boussinesq system without thermal diffusivity for the initial perturbation in Gevrey-$\frac{1}{s}$, ($1/3<s\leq 1$). The synergism of density mixing, vorticity mixing and velocity diffusion leads to the stability.
We prove the nonlinear inviscid damping for a class of monotone shear flows in $T\times [0,1]$ for initial perturbation in Gevrey-$1/s$($s>2$) class with compact support. The main idea of the proof is to use the wave operator of a slightly modified Rayleigh operator in a well chosen coordinate system.
We consider the 2D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R}$, with initial vorticity that is $δ$ close in $H^{log}_xL^2_{y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove that if $δ\ll ν^{1/2}$, where $ν$ denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time $t\gg ν^{-1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space $H^{log}_xL^2_{y}\subset L^2_{x,y}$.
We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity $Ω_{in}$ satisfies $\|Ω_{in}-(-1)\|_{H^σ}\leq εRe^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg Re^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.
In this paper, we consider the generation of magnetic island for the linearized MHD equations around the steady flowing plasma with velocity field $U_s=(u(y),0)$ and magnetic field $H_s=(b(y),0)$ in the finite channel.
In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $ν^{\frac 23+}$, here $ν$ is the viscosity coefficient.
In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.
In the paper, we establish commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains. The approach is based on Dahlberg's bilinear estimates, and the results may be regarded as an extension of [Dahlberg, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc., 97(1986), no.1, 41-48.] and [Shen, Commutator estimates for the Dirichlet-to-Neumann map in Lipschitz domains, Some topics in harmonic analysis and applications, 369C384, Adv. Lect. Math. (ALM)34, Int. Press, Somerville, MA, 2016.] to Stokes systems.