arXiv subjects

Bijun Zuo

Publications and source records attributed to Bijun Zuo.

12 recordsLinked to original sources

arXiv2025

On the positive constant in Arnold's second stability theorem for a bounded domain

For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and its vorticity $\omega$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nabla\omega/\nabla\psi 0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\bm\Lambda_1$ of a certain Laplacian eigenvalue problem. When $\nabla\omega/\nabla\psi$ reaches $\bm\Lambda_1$, instability may occur, as illustrated by a non-circular steady flow in a disk; however, a certain form of structural stability still holds. Based on these results, we establish a theorem on the rigidity and orbital stability of steady Euler flows in a disk.

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arXiv2023

Stability of degree-2 Rossby-Haurwitz waves

Rossby-Haurwitz (RH) waves are important explicit solutions of the incompressible Euler equation on a two-dimensional rotating sphere. In this paper, we prove the orbital stability of degree-2 RH waves, which confirms a conjecture proposed by A. Constantin and P. Germain in [Arch. Ration. Mech. Anal. 245, 587-644, 2022]. The proofs are based on a variational approach, with the main challenge being to establish suitable variational characterizations for the solutions under consideration. In this process, the set of rearrangements of a fixed function plays a vital role. We also apply our approach to the stability analysis of degree-1 RH waves, Arnold-type flows, and zonal flows with monotone absolute vorticity.

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arXiv2023

Stable plane Euler flows with concentrated and sign-changing vorticity

We construct a family of steady solutions to the two-dimensional incompressible Euler equation in a general bounded domain, such that the vorticity is supported in two well-separated regions of small diameter and converges to a pair of point vortices with opposite signs. Compared with previous results, we do not need to assume the existence of an isolated local minimum point of the Kirchhoff-Routh function. Moreover, due to their variational nature, the solutions obtained are Lyapunov stable in $L^p$ norm of the vorticity. The proofs are achieved by maximizing the kinetic energy over an appropriate family of rearrangement classes of sign-changing functions and studying the limiting behavior of the maximizers.

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arXiv2022

Nonlinear stability of sinusoidal Euler flows on a flat two-torus

Sinusoidal flows are an important class of explicit stationary solutions of the two-dimensional incompressible Euler equations on a flat torus. For such flows, the steam functions are eigenfunctions of the negative Laplacian. In this paper, we prove that any sinusoidal flow related to some least eigenfunction is, up to phase translations, nonlinearly stable under $L^p$ norm of the vorticity for any $1<p<+\infty$, which improves a classical stability result by Arnold based on the energy-Casimir method. The key point of the proof is to distinguish least eigenstates with fixed amplitude from others by using isovortical property of the Euler equations.

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arXiv2022

An extension of Arnold's second stability theorem in a multiply-connected domain

We give a sufficient condition for the nonlinear stability of steady flows of a two-dimensional ideal fluid in a bounded multiply-connected domain, which generalizes a stability criterion proved by Arnold in the 1960s. The most important ingredient of the proof is to establish a variational characterization for the steady flow under consideration, which is achieved based on the energy-Casimir method proposed by Arnold, and the supporting functional method introduced by Wolansky and Ghil. Nonlinear stability then follows from a compactness argument related to the variational characterization and proper use of conserved quantities of the two-dimensional Euler equations.

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arXiv2021

Global existence of weak solutions to the compressible Navier-Stokes equations with temperature-depending viscosity coefficients

This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed to be functions of the temperature, and the shear viscosity coefficient may vanish as the temperature goes to zero. The proof is to apply Galerkin method to a suitable approximate system with several parameters and obtain uniform estimates for the approximate solutions. The key ingredient in obtaining the required estimates is to apply De Giorgi's iteration to the modified temperature equation, from which we can get a lower bound for the temperature not depending on the artificial viscosity coefficient introduced in the modified momentum equation, which makes the compactness argument available as the artificial viscous term vanishes.

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arXiv2021

Location of concentrated vortices in planar steady Euler flows

In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics. The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},$ which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.

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arXiv2020

Global existence of weak solutions to the Navier-Stokes equations with temperature-depending viscosity coefficient

In this paper, the initial-boundary value problem to the three-dimensional inhomogeneous, incompressible and heat-conducting Navier-Stokes equations with temperature-depending viscosity coefficient is considered in a bounded domain. The viscosity coefficient is degenerate and may vanish in the region of absolutely zero temperature. Global existence of weak solutions to such a system is established for the large initial data. The proof is based on a three-level approximate scheme, the De Giorgi's method and compactness arguments.

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arXiv2019

On radial symmetry of rotating vortex patches in the disc

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch $D$ with angular velocity $Ω$, $Ω\geq \max\{{1}/{2},({2 l^2})/{(1-l^2)^2}\}$ or $Ω\leq -({2 l^2})/{(1-l^2)^2}$, where $l=\sup_{x\in D}|x|$, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of Gómez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

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arXiv2019

Existence of steady symmetric vortex patch in a disk

In this paper we construct a family of steady symmetric vortex patches for the incompressible Euler equations in an open disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches shrink to a given minimum point of the corresponding Kirchhoff-Routh function as the vorticity strength parameter goes to infinity.

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arXiv2019

Steady vortex patches near a rotating flow with constant vorticity in a planar bounded domain

In this paper, we study steady vortex patch solutions to the incompressible Euler equations in a planar bounded domain $D$. Let $ψ_0$ be the solution of the elliptic problem $-Δψ_{0} =1$ in $D$; $ψ_0=0$ on $\partial D$. We prove that for any finite collection of isolated maximum points of $ψ_0$, say $\{x_1,\cdot\cdot\cdot,x_k\},$ and any $k$-tuple $\vecκ=(κ_1,\cdot,\cdot,\cdot,κ_k)$ with $κ_i>0$ and $|\vecκ|:=\sum_{i=1}^kκ_i<<1,$ there exists a steady solution of the Euler equations such that the vorticity has the form $ω^{\vecκ}=1-I_{\cup_{i=1}^k A^{\vecκ}_i}$, where $I$ denotes the characteristic function, $|A^{\vecκ}_i|=κ_i$ and $A^{\vecκ}_i$ "shrinks" to $x_i$ as $|\vecκ|\to 0$.

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arXiv2018

Energy and cross-helicity conservation for the three-dimensional ideal MHD equations in bounded domain

In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in \cite{Caflisch 97}, and the boundedness and the Besov-type continuity for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that the Besov-type continuity condition near the boundary is consistent with the interior Besov regularity, which is a new condition we proposed in the present paper.

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