arXiv subjects

Dongfen Bian

Publications and source records attributed to Dongfen Bian.

31 records · Page 2Linked to original sources

arXiv2022

Long waves instabilities

The aim of this paper is to give a detailed presentation of long wave instabilities of shear layers for Navier Stokes equations, and in particular to give a simple and easy to read presentation of the study of Orr Sommerfeld equation and to detail the analysis of its adjoint. Using these analyses we prove the existence of long wave instabilities in the case of slowly rotating fluids, slightly compressible fluids and for Navier boundary conditions, under smallness conditions.

math.AP↗

arXiv2022

Stability threshold for 2D shear flows near Couette of the Navier-Stokes equation

In this paper, we consider the stability threshold of the 2D shear flow $(U(y),0)^{\top}$ of the Navier-Stokes equation at high Reynolds number $Re$. When the shear flow is near in Sobolev norm to the Couette flow $(y,0)^{\top}$ in some sense, we prove that if the initial data $u_0$ satisfies $\|u_0-(U(y),0)^{\top}\|\leq εRe^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to the Couette flow for $t\gg Re^{1/3}$, as $t\to\infty$.

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arXiv2020

Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field

In this paper, we consider the Boussinesq equations with magnetohydrodynamics convection in the domain $\mathbb{T} \times \mathbb{R}$ and establishes the nonlinear stability of the Couette flow $(\mathbf{u}_{sh} = (y,0), \mathbf{b}_{sh} = (1,0), p_{sh} = 0, θ_{sh} = 0$). The novelty in this paper is that we design a new Fourier multiplier operator by using the properties of the enhanced dissipation to overcome the difficult term $\partial_{xy}(-Δ)^{-1}j$ in the linearized and nonlinear system. Then, we prove the asymptotic stability for the linearized system. Finally, we establish the nonlinear stability for the full system by bootstrap principle.

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arXiv2020

Stability threshold for 2D shear flows of the Boussinesq system near Couette

In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain $\mathbb{T} \times \mathbb{R}$. The main goal is to prove the nonlinear stability of the shear flow $(U^S,Θ^S)=((e^{νt\partial_{yy}}U(y),0)^{\top},αy)$ with $U(y)$ close to $y$ and $α\geq0$. We separate two cases: one is $α\geq 0$ small scaling with the viscosity coefficients and the case without smallness of $α$ and fixed heat diffusion coefficient. The novelty here is that we don't require $μ=ν$ and only need to assume that $μ$ is scaled with $ν$ or fixed, where $μ$ is the inverse of the Reynolds number and $ν$ is the heat diffusion coefficient.

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arXiv2020

Linear instability of $Z$-pinch in plasma (I): Inviscid case

The z-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the z direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that any such a z-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system.

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arXiv2020

Linear instability of $Z$-pinch in plasma (II): Viscous case

The z-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the $z$ direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such z-pinch is always unstable. We also establish the existence of a largest growing mode, which dominates the linear growth of the linear MHD system.

math.AP↗

arXiv2019

Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball

In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(ρ, u, θ)$, in the class $C^1([0,T]; H^m(Ω))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauchy probelm is considered, to the case that with physical boundary.

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arXiv2017

Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects

This paper is concerned with the initial-boundary value problem to 2D magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. First, we establish the global weak solutions under the minimal initial assumption. Then by imposing higher regularity assumption on the initial data, we obtain the global strong solution with uniqueness. Moreover, the exponential decay estimate of the solution is obtained.

math.AP↗

arXiv2017

Global solution to the nematic liquid crystal flows with heat effect

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain $Ω\subset\mathbb{R}^N$ ($N=2, 3$) are studied in this paper. Following Danchin's method in [J. Math. Fluid Mech., 2006], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu-Wang's result in [Commun. Math. Phys., 2010] to the non-isothermal case.

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arXiv2015

Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, F. M. Huang, A. Matsumura and X. D. Shi showed that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and the key point is to deduce the desired uniform positive lower and upper bounds on the density.

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arXiv2012

Global well-posedness for the two dimensional compressible MHD equations with large data

In this paper we are concerned with the global well-posedness for the compressible MHD equations with large data. We show that if the shear viscosity $μ$ is a positive constant and the bulk viscosity $λ$ is the power function of the density, that is, $λ(ρ)=ρ^β$ with $β>3$, then the two dimensional compressible MHD system with the periodic boundary conditions on the torus $\mathbb{T}^2$ have a unique global classical solution $(ρ, u,H)$. In this work we extended the results about compressible Navier-Stokes equations in \cite{Karzhikhov} to compressible MHD equations by applying several new techniques to overcome the coupling between velocity and magnetic field.

math.AP↗

arXiv2012

Optimal convergence rates for the three-dimensional turbulent flow equations

In this paper we are concerned with the convergence rate of solutions to the three-dimensional turbulent flow equations. By combining the $L^p$-$L^q$ estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space, when the initial perturbation of the equilibrium state is small in $H^3$-framework. More precisely, the optimal convergence rates of the solutions and its first order derivatives in $L^2$-norm are obtained when the $L^p$-norm of the perturbation is bounded for some $p\in[1, 6/5)$.

math.AP↗

arXiv2012

Global existence of smooth solutions to three-dimensional turbulent flow equations

In this paper we are concerned with the global existence of smooth solutions to the turbulent flow equations for compressible flows in $\mathbb{R}^3$. The global well-posedness is proved under the condition that the initial data are close to the standard equilibrium state in $H^3$-framework. The proof relies on energy estimates about velocity, temperature, turbulent kinetic energy and rate of viscous dissipation. We use several new techniques to overcome the difficulties from the product of two functions and higher order norms. This is the first result concerning $k$-$\varepsilon$ model equations.

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