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We study the asymptotic linear stability of a two-parameter family of solitary waves for the isothermal Euler–Poisson system. When the linearized equations about the solitary waves are considered, the associated eigenvalue problem in $$L^2$$ space has a zero eigenvalue embedded in the neutral spectrum, i.e., there is no spectral gap. To resolve this issue, use is made of an exponentially weighted $$L^2$$ norm so that the essential spectrum is strictly shifted into the left-half plane, and this is closely related to the fact that solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain long-wavelength scaling, we show that the Evans function for the Euler–Poisson system converges to that for the Korteweg–de Vries (KdV) equation as an amplitude parameter tends to zero, from which we deduce that the origin is the only eigenvalue on its natural domain with algebraic multiplicity two. We also show that the solitary waves are spectrally stable in $$L^2$$ space. Moreover, we discuss (in)stability of large amplitude solitary waves.

This paper devotes to the existence of an N N -dimensional inertial manifold for a class of singularly, i.e. A ( t ) A(t) may degenerate to 0 0 at some time t t , non-autonomous parabolic equations ∂ t u + A ( t ) u = F ( t , u ) + g ( x , t ) , t > τ ; u | t = τ = u τ ( x ) , x ∈ Ω , \begin{equation*} \partial _{t}u+A(t)u=F(t,u)+g(x,t),\;t>\tau ;\; \; u|_{t=\tau }=u_{\tau }(x),\;x\in \Omega , \end{equation*} where A ( t ) ≥ 0 A(t)\geq 0 for any t ≥ τ t\geq \tau , and Ω ⊂ R d \Omega \subset \mathbb {R}^{d} is a bounded domain with smooth boundary. Since the operator A ( t ) A(t) may degenerate, a compatibility condition for the operator A ( t ) A(t) and the nonlinear term F ( t , u ) F(t,u) was proposed to construct the inertial manifolds.

In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov-Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in $H^2(\mathbb R^2)$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in [Dohnal and D\"orfler, Multiscale Model. Simul., p. 162-191, 11 (2013)]., Comment: Improved version in accordance to the Referees' suggestions. In particular Lemma 4.3 rectified. Main results unchanged

We construct "almost periodic'' unbounded domains, where a large class of elliptic spectral problems have essential spectra possessing peculiar structure: they consist of monotone, non-negative sequences of isolated points and thus have infinitely many gaps.

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier–Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.

We give a two sided estimate on the spectral gap for the Boltzmann measures μh on the circle. We prove that the spectral gap is greater than 1 for any h ∈ ℝ and the spectral gap tends to the positive infinity as h → ∞ with speed ∣h∣.

A model for the spectrum of the lateral velocity component $$S_v(f)$$ is developed for a frequency range from about 0.2 $$\hbox {day}^{-1}$$ to the turbulence inertial subrange, with the intent of improving the calculation of flow meandering over areas the size of offshore wind farms and clusters. These sizes can correspond to a temporal scale of several hours, much larger than the validity limit of typical boundary-layer models, such as the Kaimal model, or the Mikkelsen–Tchen model. The development of the model is based on observations from one site and verified with observations from another site up to a height of 241 m. The model describes three ranges: (1) the mesoscale from $$0.2\ \hbox {day}^{-1}$$ to about $$10^{-3}\ \hbox {Hz}$$ where a mesoscale spectral model from Larsen et al. (2013: QJR Meteorol Soc 139: 685–700) is used; (2) the spectral gap where the normalized v spectrum, $$fS_v(f)$$ , can be approximated to be a constant; (3) the high-frequency range where a boundary-layer model is used. In order to demonstrate a general applicability of the lateral velocity spectrum model to reproduce the statistics of wind-direction variability, models for both horizontal velocity components, u and v, are used through an inverse Fourier transform technique to produce time series of both components, which in theory could have been the ensemble for calculating the corresponding spectra. The ensemble is then used to calculate directional statistics, which in turn are compared with corresponding statistics from the measured time series. We demonstrate the relevance of the constructed spectral models for calculating meandering effects for large wind farms and wind-farm clusters.

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained., Comment: 38 pages

The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.

This paper deals with the asymptotic behavior of solution to the linearized ellipsoidal BGK model in torus. We prove that the solution converges exponentially to the equilibrium in the weighted Sobolev spaces with polynomial weight. Our exponential decay rate $$e^{-\lambda t}$$ is optimal in the sense that $$\lambda >0$$ equals to the spectral gap of the linearized operator in the standard Hilbert space. Our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference Hilbert space, the factorization method, and the enlargement of the functional space for the associated semigroup.

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

We study a $1$-dimensional chain of $N$ weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with $N$ dependent bounds) perturbations of the harmonic ones. We show how a generalized version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by F. Baudoin (2017). By that we prove exponential convergence to non-equilibrium steady state (NESS) in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $N^{-3}$. For the purely harmonic chain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in this set up, the spectral gap decays at most polynomially with $N$., Comment: 54 pages, 1 figure, comments are welcome

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown., Comment: 13 pages

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $$L^2$$ L 2 -norm for smooth flows in the pre-asymptotic high Reynolds number regime.

The asymptotics of eigenvalues appearing near the lower edge of a spectral gap of the Dirichlet problem is studied for the Laplace operator in a d-dimensional periodic waveguide with a singular perturbation of the boundary by creating a hole with a small diameter e. Several versions of the structure of the gap edge are considered. As usual, the asymptotic formulas are different in the cases d ≥ 3 and d = 2, where the eigenvalues occur at distances O(e2(d−2)) or O(e2d) and O(|ln e|−2) or O(e4), respectively, from the gap edge. Other types of singular perturbation of the waveguide surface and other types of boundary conditions are discussed, which provide the appearance of eigenvalues near both edges of one or several gaps.

This article concerns the Hamiltonian elliptic system: $$ \textstyle\begin{cases} -\Delta \varphi +V(x)\varphi =G_{\psi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ -\Delta \psi +V(x)\psi =G_{\varphi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ \varphi , \psi \in H^{1}(\mathbb {R}^{N}). \end{cases} $$ { − Δ φ + V ( x ) φ = G ψ ( x , φ , ψ ) in R N , − Δ ψ + V ( x ) ψ = G φ ( x , φ , ψ ) in R N , φ , ψ ∈ H 1 ( R N ) . Assuming that the potential V is periodic and 0 lies in a spectral gap of $\sigma (-\Delta +V)$ σ ( − Δ + V ) , least energy solution of the system is obtained for the super-quadratic case with a new technical condition, and the existence of ground state solutions of Nehari–Pankov type is established for the asymptotically quadratic case. The results obtained in the paper generalize and improve related ones in the literature.

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied. Bibliography: 38 titles.

We study the simultaneous exact controllability of two vibrating strings with variable physical coefficients and controlled from a common endpoint. We give sufficient conditions on the physical coefficients for which the eigenfrequencies of both systems do not coincide and the associated spectral gap is uniformly positive. Under these conditions, we show that these systems are simultaneously exactly controllable.

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices $H=\langle H\rangle +\delta H$, with $\langle H\rangle$ rank-$1$ and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement., Comment: 20 pages, 4 figures. Clarifications and appendices added

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ \epsilon$ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are $\epsilon$-approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales., Comment: 37 pages, 6 figures

A notion of standard vertex conditions for beam operarators (the fourth derivative) on metric graphs is presented, and the spectral gap (the difference between the first two eigenvalues) for the operator with these conditions is studied. Upper and lower estimates for the spectral gap are obtained, and it is shown that stronger estimates can be obtained for certain classes of graphs. Graph surgery is used as a technique for estimation. A geometric version of the Ambartsumian theorem for networks of beams is proved.

Global existence of the Euler alignment system in dimension 2 and higher is only partially resolved at the moment. In the smooth kernel case, no sharp threshold condition is known, and in the singular case, the question is wide open. However, certain classes of special solutions do permit global existence. Those include solutions with small initial velocity fluctuations, small spectral gap of the symmetric velocity strain tensor, or special classes respecting symmetries of the system. This chapter presents a collection of such multidimensional resularity results along with relevant analytical techniques.

This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpinski gasket. The main result shows the remarkable fact that this minimal gap is achieved and coincides with the spectral gap. The Dirichlet case is more challenging and requires some key observations in the behavior of the dynamical system that describes the spectrum., Comment: 14 pages

International audience; By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

In this paper, we study the second-order Hamiltonian systems $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$ u ¨ − L ( t ) u + ∇ W ( t , u ) = 0 , where $t\in \mathbb{R}$ t ∈ R , $u\in \mathbb{R}^{N}$ u ∈ R N , L and W depend periodically on t, 0 lies in a spectral gap of the operator $-d^{2}/dt^{2}+L(t)$ − d 2 / d t 2 + L ( t ) and $W(t,x)$ W ( t , x ) is locally superquadratic. Replacing the common superquadratic condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ uniformly in $t\in \mathbb{R}$ t ∈ R by the local condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ a.e. $t\in J$ t ∈ J for some open interval $J\subset \mathbb{R}$ J ⊂ R , we prove the existence of one nontrivial homoclinic soluiton for the above problem.

The work of the first and third authors was supported by the project MTM2017- 85067-P, funded by the Spanish government and the European Regional Development Fund and they gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory. The work of the second author was supported by the ANR project NOLO, grant ANR-20-CE40-0015, funded by the French Ministry of Research. The work of the third author was also supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-0718, by the ``la Caixa"" Foundation, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme grant 639638., We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation., BERC, French Ministry of Research, Hausdorff Research Institute for Mathematics ANR-20-CE40-0015, Spanish government, European Commission 639638 EC, European Research Council, Ministerio de Economía y Competitividad ¡SEV-2017-0718 MINECO, European Regional Development Fund

For gradient flows of energies, both spectral renormalization (SRN) and energy landscape (EL) techniques have been used to establish slow motion of orbits near low-energy manifold. We show that both methods are applicable to flows induced by families of gradients and compare the scope and specificity of the results. The SRN techniques capture the flow in a thinner neighborhood of the manifold, affording a leading order representation of the slow flow via as projection of the flow onto the tangent plane of the manifold. The SRN approach requires a spectral gap in the linearization of the full gradient flow about the points on the low-energy manifold. We provide conditions on the choice of gradient under which the spectral gap is preserved, and show that up to reparameterization the slow flow is invariant under these choices of gradients. The EL methods estimate the magnitude of the slow flow, but cannot capture its leading order form. However the EL only requires normal coercivity for the second variation of the energy, and does not require spectral conditions on the linearization of the full flow. It thus applies to a much larger class of gradients of a given energy. We develop conditions under which the assumptions of the SRN method imply the applicability of the EL method, and identify a large family of gradients for which the EL methods apply. In particular we apply both approaches to derive the interaction of multi-pulse solutions within the 1+1D Functionalized Cahn-Hilliard (FCH) gradient flow, deriving gradient invariance for a class of gradients arising from powers of a homogeneous differential operator., Comment: 28 pages

We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral-gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time., Comment: 19 pages. Some typos were fixed in this version

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer operator of the same speed than entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of H{\"o}lder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situation, including Pomeau-Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results., Comment: v3: The published article (Ergodic Theory Dynam. Systems 40, 2020) contained a significant error in Lemma 2.14, used in the core Theorem 4.1. This is a consolidated version of the article, with the error corrected (and a few other minor points improved along the way). In order to fix the error, the assumption on coupling in 4.1 needs to be slightly modified (see also Definition 2.12) and Lemma 2.14 (now numbered 2.15) has been adjusted. Section 5.1 provides a criterion to ensure this new hypothesis in our cases of interest, so that all other results are unaffected. I apologize to readers of the previous version for this embarrassing mistake, and warmly thank Manuel Stadlbauer for pointing out this error to me

This work deals with the Cauchy problem and the asymptotic behavior of the solution of the fermion equation in the Sobolev spaces with a polynomial weight in the torus. We first investigate the linearized equation and obtain the optimal exponential decay rate for the associated semigroup. Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space, the factorization method and the enlargement of the functional space. We then turn to the nonlinear equation and prove the global existence and uniqueness of solutions in a close-to-equilibrium regime. Moreover, we prove an exponential stability for such a solution with the optimal decay rate given by the semigroup decay of the linearized equation.

In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L 2 ( μ − 1 2 ) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017).

The spectral element method (SEM) is a mimetic finite element method with several properties that make it a desirable choice for numerical modeling. Although the linear dispersion properties of this method have been analyzed extensively for the case of the 1D inviscid advection equation, practical implementations of the SEM frequently employ hyperdiffusion for stabilization. As argued in this paper, hyperdiffusion has a pronounced impact on the accuracy of the discrete wave modes and the dispersive properties of the SEM. When applied with an appropriately large coefficient, hyperdiffusion is effective at removing the spectral gap and improving the stability of the 1D advection equation. This study also considers the SEM as applied to the 2D linearized shallow-water equations, where hyperdiffusion in the form of scalar diffusion, divergence damping, and vorticity damping are analyzed. To the extent possible, guidance on the choice of hyperdiffusion coefficients is provided. A brief discussion of the comparative impact of local element filtering is included.

Let $f$ be a zero-mean continuous stationary Gaussian process on ${\mathbb R}$ whose spectral measure vanishes in a $\delta$-neighborhood of the origin. Then the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-c\delta^2 L^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions., Comment: 15 pages. To appear in IMRN (Inter. Math. Res. Notices)

We establish a new method for obtaining nonconvex spectral enclosures for operators associated with second‐order differential equations z:(t) + Dz(t) + A₀z(t) = 0 in a Hilbert space. In particular, we succeed in establishing the existence of a spectral gap, which is the first result of this kind since the seminal results of Krein and Langer for oscillations of damped systems. While the latter and other spectral bounds are confined to dampings D that are symmetric and dominated by A0, we allow for accretive D of equal strength as A₀. To achieve these results, we prove new abstract spectral inclusion results that are much more powerful than classical numerical range bounds. Two different applications, small transverse oscillations of a horizontal pipe carrying a steady‐state flow of an ideal incompressible fluid and wave equations with strong (viscoelastic and frictional) damping, illustrate that our new bounds are explicit.

We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Kantorovich-Wasserstein distance. Furthermore, if one of the densities is Gaussian and the other is log-concave, or more generally has positive spectral gap, then the deficit in the EPI can be controlled in terms of the $L^{1}$ -Kantorovich-Wasserstein distance or relative entropy, respectively. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Kantorovich-Wasserstein distance when densities are uniformly log-concave on sets of measure arbitrarily close to one. Our stability results can be extended to non-log-concave densities, provided certain regularity conditions are met. The proofs are based on mass transportation.