Semigroup Estimates Research Articles

Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity

In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation having a degenerate diffusion at spatial infinity and weighted energy estimates for the damped wave equation. To construct a suitable weight function for the energy estimates, we study a certain elliptic problem.

Besov-weak-Herz spaces and global solutions for Navier–Stokes equations

We consider the incompressible Navier-stokes equations (NS) in $\mathbb{R}^{n}$ for $n\geq2$. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These spaces are larger than some critical spaces considered in previous works for (NS). For our purposes, we need to develop a basic theory for BWH-spaces containing properties and estimates such as heat semigroup estimates, embedding theorems, interpolation properties, among others. In particular, it is proved a characterization of Besov-weak-Herz spaces as interpolation of Sobolev-weak-Herz ones, which is key in our arguments. Self-similarity and asymptotic behavior of solutions are also discussed. Our class of spaces and its properties developed here could also be employed to study other PDEs of elliptic, parabolic and conservation-law type.

Uniform gradient estimates on manifolds with a boundary and applications

We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux’s argument, and uniform quantitative gradient estimates, firstly for $$C^2_b$$ functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians.

Gradient estimates for SDEs without monotonicity type conditions

We prove gradient estimates for transition Markov semigroups (Pt) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C1-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for DxPtφ of the formt|DxPtφ(x)|≤c(1+|x|k)‖φ‖∞,t∈(0,1], φ∈Cb(Rd), x∈Rd. We use two main tools. First, we consider a Feynman–Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut–Elworthy–Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for DxPtφ without weights do not hold.

Asymptotic behaviour of the solutions to a virus dynamics model with diffusion

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the $L^1$ boundedness of the orbit. Then the pre-compactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number $R_0$ is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if $R_0>1$, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

Couplings, gradient estimates and logarithmic Sobolev inequalitiy for Langevin bridges

In this paper we establish quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes. They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, bounds of the distance between bridges of different Langevin dynamics, and a logarithmic Sobolev inequality for bridge measures. The existence of an invariant measure for the bridges is also discussed and quantitative bounds for the convergence to the invariant measure are proven. All results are based on a seemingly new expression of the drift of a bridge in terms of the reciprocal characteristic, which, roughly speaking, quantifies the “mean acceleration” of a bridge.

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L 0 + V be a Schrodinger type operator, where L 0 is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L q boundedness of Riesz transform ∇ m L −1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e −t L . Furthermore, the authors impose extra regularity assumptions on V to obtain the L q boundedness of Riesz transform ∇ m L −1/2 for some q > 2. In particular, these results are applied to the more interesting Schrodinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.

Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers

AbstractIn this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates

Let $(X, d, \mu)$ be a space of homogeneous type equipped with a distance $d$ and a measure $\mu$. Assume that $L$ is a closed linear operator which generates an analytic semigroup $e^{-tL}, t > 0$. Also assume that $L$ has a bounded $H_\infty$-calculus on $L^2(X)$ and satisfies the $L^p-L^q$ semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel bounds). The aim of this paper is to establish a theory of inhomogeneous Besov spaces associated to such an operator $L$. We prove the molecular decompositions for the new Besov spaces and obtain the boundedness of the fractional powers $(I+L)^{-\gamma}, \gamma > 0$ on these Besov spaces. Finally, we carry out a comparison between our new Besov spaces and the classical Besov spaces.

Semigroup estimates and fast-slow dynamics in parabolic-hyperbolic systems

Abstract We present a general procedure to describe slow dynamics in parabolic-hyperbolic systems, under suitable assumptions on the terms appearing in the equations. In particular, our strategy relies in semigroup estimates for the evolution system associated to the family of time-dependent linearized operators around an interfaced configuration. We finally apply the general strategy to the explicit example of an hyperbolic system with relaxation.

Large time behavior of solutions for degenerate p-degree Fisher equation with algebraic decaying initial data

This paper is concerned with the large time behavior of solutions to the one-dimensional degenerate p-degree Fisher equation, where the initial data are assumed to be asymptotically front-like and to decay to zero non-exponentially at one end. By applying sub–super solution method we first prove the Lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space. Further, by using semigroup estimates, sub–super solution method and the nonlinear stability of the traveling waves, we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite, which is solely determined by the detailed decaying rate of the initial data.

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.

Sharp estimates of unimodular multipliers on frequency decomposition spaces

This paper is devoted to give a general method for the study of unimodular multipliers on certain function spaces defined by decomposition on the frequency plane. These spaces include modulation spaces, α-modulation spaces and (homogeneous and inhomogeneous) Besov spaces. We give a complete characterization of the Fourier multipliers on these function spaces, and also give a characterization of unimodular multipliers under some mild assumptions. As applications, we obtain some sharp boundedness properties of unimodular Fourier multipliers between these function spaces. We also obtain the asymptotic estimates for certain free dispersive semigroups.

Large time behavior of solutions to Vlasov-Poisson-Landau\\ (Fokker-Planck) equations

In this paper, we consider the initial value problems for Vlasov-Poisson-Fokker-Planck (VPFP) equations and Vlasov-Poisson-Landau (VPL) equations. We give the spectrum analysis and semigroup estimates on the linearized VPFP equations and VPL equations around their equilibrium states and show the optimal convergence rates of global solution to nonlinear problems. We show that the solution to VPFP equations tends to the equilibrium state at the exponential convergence rate, the solution to VPL equations tends to the equilibrium state at the algebraic convergence rate $(1+t)^{-1/4}$ when initial values are small perturbations of global Maxwellian.

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

We prove the multidimensional stability of planar traveling waves for scalar nonlocal Allen-Cahn equations using semigroup estimates. We show that if the traveling wave is spectrally stable in one space dimension, then it is stable in $n$-space dimension, $n\geq 2$, with perturbations of the traveling wave decaying like $t^{-(n-1)/4}$ as $t\rightarrow +\infty$ in $H^k(\mathbb{R}^n)$ for $k\geq \left[\frac{n+1}{2}\right]$.

Perturbations of positive semigroups on $$L_p$$ L p -spaces

We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) \(L_p\)-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for the respective resolvents. The results have applications to kernel estimates for semigroups induced by accretive and non-local forms on \(\sigma \)-finite measure spaces.

Linear Stability for Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

We consider linear stability for planar transition front solutions \(\bar{u} (x_1)\) arising in multidimensional (i.e., \(x\in \mathbb {R}^n\)) Cahn–Hilliard systems. In previous work the author has established that the linear operator obtained from linearization about the transition front has (after Fourier transform in the transverse variable \(\tilde{x} = (x_2, x_3, \dots , x_n) \mapsto \xi \)) a leading eigenvalue that moves into the stable (Re \({\uplambda }< 0\)) half-plane at rate \(|\xi |^3\). This constitutes precisely the type of borderline case that has been effectively analyzed by the pointwise semigroup methods of Zumbrun and Howard, and we follow that approach here. In particular, the approach can be viewed as a three-step process including: (1) characterization of the spectrum of the linearized operator; (2) derivation of linear semigroup estimates, typically encoded into estimates on an appropriate Green’s function; and (3) implementation of an iterative process to accommodate nonlinearities. In this paper we address Step (2) in the case of multidimensional Cahn–Hilliard systems.

L 1-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

We investigate selfadjoint positivity preserving $C_0$-semigroups that are dominated by the free heat semigroup on $\mathbb R^d$. Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schr\"odinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum we prove estimates of their $L_1$-norm in terms of the $L_2$-norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

The Leray-Gårding method for finite difference schemes

In [Ler53] and [ Gar56], Leray and Garding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, Gar56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu [Tre84, KW93].

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotactic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.