We consider time-independent solutions of hyperbolic equations such as $\d_{tt}u -Δu= f(x,u)$ where $f$ is convex in $u$. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as $\d_t u -Δu= f(x,u)$. Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.
In this work we prove the nonlinear instability of inhomogeneous steady states solutions to the Hamiltonian Mean Field (HMF) model. We first study the linear instability of this model under a simple criterion by adapting the techniques developed in [19]. In a second part, we extend to the inhomogeneous case some techniques developed in [14, 17, 18] and prove a nonlinear instability result under the same criterion.
We investigate the instability and stability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $Ω$ of class $C^2$. When the steady density is heavier with increasing height (i.e., the Rayleigh-Taylor steady-state), we show that the steady-state is linear unstable (i.e., the linear solution grows in time in $H^2$) by constructing a (standard) energy functional and exploiting the modified variational method. Then, by introducing a new energy functional and using a careful bootstrap argument, we further show that the stea…
We show that a steady-state solution ${\bf U}$ to the system of equations of a Navier-Stokes flow past a rotating body is nonlinearly unstable if the associated linear operator $\cal L$ has a part of the spectrum in the half-plane $\{λ\in\mathbb C;\ {\rm Re}\, λ>0\}$. Our result does not follow from known methods, %on spectral instability, mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied. As an auxiliary result of independent interest, we also show that the uniform growth bound of the $C_0$--semigroup ${\rm e}^{{\cal L} t}$ is equ…
We study the steady uniphase and multiphase solutions of the discretized nonlinear damped wave equation.Conditions for the stability abd instability of the steady solutions are given;in the instability case the linear stable and unstable associated manifolds are described.
For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C-smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the C^1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the p…
In [F. Jiang, S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014) 831--863], Jiang et.al. investigated the instability of Rayleigh--Taylor steady-state of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $Ω$ of class $C^2$. In particular, they proved the steady-state is nonlinearly unstable under a restrictive condition of that the derivative function of steady density possesses a positive lower bound. In this article, by exploiting a standard energy functional a…
We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.
We consider a boundary-value problem describing the steady motion of a two-component mixture of viscous compressible heat-conducting fluids in a bounded domain. We make no simplifying assumptions except for postulating the coincidence of phase temperatures (which is physically justified in certain situations), that is, we retain all summands in equations that are a natural generalization of the Navier-Stokes-Fourier model of the motion of a one-component medium. We prove the existence of weak generalized solutions of the problem.
We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized…
We show that the elliptic problem $Δu+f(u)=0$ in $\mathbb{R}^N$, $N\geq 1$, with $f\in C^1(\mathbb{R})$ and $f(0)=0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is signchanging. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of mono…
In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fxed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained stea…
In this paper, we study the diffusive limit of the steady state radiative heat transfer system for non-homogeneous Dirichlet boundary conditions in a bounded domain with flat boundaries. A composite approximate solution is constructed using asymptotic analysis taking into account of the boundary layers. The convergence to the approximate solution in the diffusive limit is proved using a Banach fixed point theorem. The major difficulty lies on the nonlinear coupling between elliptic and kinetic transport equations. To overcome this problem, a spectral assumption ensuring the linear stability of…
In this note we revisit the classical subject of the Rayleigh-Taylor instability in presence of an incompressible background shear flow. We derive a formula for the essential spectral radius of the evolution group generated by the linearization near the steady state and reveal that the velocity variations neutralize shortwave instabilities. The formula is a direct generalization of the result of H. J. Hwang and Y. Guo in the hydrostatic case \cite{HG}. Furthermore, we construct a class of steady states which posses unstable discrete spectrum with neutral essential spectrum. The technique invol…
In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number $R=O(10^3)$. This problem is modelled as a MHD flow bounded between lateral walls under slip wall boundary condition. The existence of the secondary steady state flow is now proved. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at $R=O(10^3)$. Further transition of the secondary flow is observed numerically. Especially, well developed turbul…
We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result fo…
We prove that there exists a strong solution to the Dirichlet boundary value problem for the steady Navier-Stokes equations of a compressible heat-conductive fluid with large external forces in a bounded domain $R^d (d = 2, 3)$, provided that the Mach number is appropriately small. At the same time, the low Mach number limit is rigorously verified. The basic idea in the proof is to split the equations into two parts, one of which is similar to the steady incompressible Navier-Stokes equations with large forces, while another part corresponds to the steady compressible heat-conductive Navier-St…
We investigate the thermal instability of a smooth equilibrium state, in which the density function satisfies Schwarzschild's (instability) condition, to a compressible heat-conducting viscous flow without heat conductivity in the presence of a uniform gravitational field in a three-dimensional bounded domain. We show that the equilibrium state is linearly unstable by a modified variational method. Then, based on the constructed linearly unstable solutions and a local well-posedness result of classical solutions to the original nonlinear problem, we further construct the initial data of linear…
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $κ>0$. We prove their linear instability when $κ$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability propo…
The dynamics of gaseous stars can be described by the Euler-Poisson system. Inspired by Rein's stability result for $γ>{4/3}$, we prove the nonlinear instability of steady states for the adiabatic exponent $γ={6/5}$ in spherically symmetric and isentropic motion.
We study instability of unidirectional flows for the linearized 2D Navier-Stokes equations on the torus. Unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector $\mathbf p \in \mathbb Z^{2}$. Using Fourier series and a geometric decomposition allows us to decompose the linearized operator $L_{B}$ acting on the space $\ell^{2}(\mathbb Z^{2})$ about this steady state as a direct sum of linear operators $L_{B,\mathbf q}$ acting on $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$. Using the method of continued fr…
