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We investigate the dynamics of roll solutions at the zigzag boundary of the planar Swift-Hohenberg equation. Linear analysis shows an algebraic decay of small perturbation with a $t^{- 1/4}$ rate, instead of the classical $t^{- 1/2}$ diffusive decay rate, due to the degeneracy of the quadratic term of the continuation of the translational mode of the linearized operator in the Bloch-Fourier spaces. The proof is based on a decomposition of the neutral mode and the faster decaying modes in the Bloch-Fourier space, and a fixed-point argument, demonstrating the irrelevancy of the nonlinear terms.
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For a domain $Omega subset mathbb{R}^n$ and a small number $frak{T} > 0$, let [ mathcal{E}_0(Omega) = lambda_1(Omega) + {frak{T}} {text{tor}}(Omega) = inf_{u, w in H^1_0(Omega)setminus {0}} frac{int | abla u|^2}{int u^2} + {frak{T}} int frac{1}{2} | abla w|^2 - w ] be a modification of the first Dirichlet eigenvalue of $Omega$. It is well-known that over all $Omega$ with a given volume, the only sets attaining the infimum of $mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $partial B_R$ with bounded $C^2$ norm, [ int |u_{Omega} - u_{B_R}|^2 + |Omega triangle B_R|^2 leq C [mathcal{E}_0(Omega) - mathcal{E}_0(B_R)] ] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.
We study the stability of traveling waves of nonlinear Schrodinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critical nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For traveling waves without vortices (i.e. nonvanishing) of general nonlinearity in any dimension, we find the sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable and find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of above cases, we construct unstable and stable invariant manifolds.
This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium at an identified quantitative rate.
We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
We exploit the existence and nonlinear stability of boundary spike/layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero. Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation, to a nonlinear nonlocality which is resolved by the technique of taking antiderivatives, i.e. working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardys inequality that fully captures the dissipative structure of the system.
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