Resolvent Estimates Research Articles

Nonlocal Fractional Differential Equations and Applications

The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional differential equations are studied.

Semiclassical resolvent bound for compactly supporte $L^\infty$ potentials

We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension n \ge 1 . We require the potential belong to L^\infty(\mathbb{R}^n) and have compact support, but do not require that it have distributional derivatives in L^\infty(\mathbb{R}^n) . The weighted resolvent norm is bounded by e^{Ch^{-4/3}\log(h^{-1})} , where h is the semiclassical parameter.

Transition Threshold for the 2-D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

Absence of eigenvalues of non‐self‐adjoint Robin Laplacians on the half‐space

By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self-adjoint setting, where we obtain quantum-mechanically natural conditions.

Positive Semigroups Using Resolvent Estimate Bounds on Sharp Growth Rates

In this paper, we study growth rates for strongly continuous semigroups. We fixate that a growth rate for the resolvent estimate on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroupis positive and the underlying space is an -space or a space of continuous functions. Also proved variations of the main results on fractional domains; these are valid on more general Banach spaces by Jan Rozendaal and Mark Veraar. In the second part apply the main theorem to prove optimality in a classical example of a perturbed wave equation which shows unusual sequence of spectral behavior.

Uniform Resolvent Estimates on Manifolds of Bounded Curvature

We establish $$L^{q*}\rightarrow L^q$$ bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of $$[0,\infty )$$, the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.

Enclosure of the numerical range and resolvent estimates of non-selfadjoint operator functions

In this paper we discuss the relationship between the numerical range of an extensive class of unbounded operator functions and the joint numerical range of the operator coefficients. Furthermore, we derive methods on how to find estimates of the joint numerical range. Those estimates are used to obtain explicitly computable enclosures of the numerical range of the operator function and resolvent estimates. The enclosure and upper estimate of the norm of the resolvent are optimal given the estimate of the joint numerical range.

High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general “worst case” bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle.

Robust Output Regulation of a Flexible Satellite

We consider a PDE-ODE model of a satellite and robust output regulation of the corresponding model. The satellite is composed of two flexible solar panels and a rigid center body. Exponential stability of the model is proved using passivity and resolvent estimates in the port-Hamiltonian framework. In addition, we construct a simple low-gain controller for robust output regulation of the satellite model.

Decays for Kelvin--Voigt Damped Wave Equations I: The Black Box Perturbative Method

We show in this article how perturbative approaches from N. Burq and M. Hitrik [Math. Res. Lett., 14 (2007), pp. 35--47] and the black box strategy from N. Burq and M. Zworski [J. Amer. Math. Soc., 17 (2004), pp. 443--471] allow us to obtain decay rates for Kelvin--Voigt damped wave equations from quite standard resolvent estimates: Carleman estimates or geometric control estimates for Helmoltz equation; Carleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin--Voigt damping, such an approach is unlikely to allow for the optimal results when additional geometric assumptions are considered, it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori).

The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit

Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let V denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator A_h = −h^2 ∆ + i V in the semi-classical limit h → 0. We obtain both the asymptotic behaviour of the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for potentials for which the set where the current (or ∇V) is normal to the boundary is discrete, in contrast with the present case where V is constant along the conducting surfaces.

Optimal constants in nontrapping resolvent estimates and applications in numerical analysis

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ times the length of the longest generalized bicharacteristic of $|\xi|_g^2-1$ remaining in the support of $\chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.

Spectral inequality and resolvent estimate for the bi-Laplace operator

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called “clamped” boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously.We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp (C \mu^{1/4}) where \mu is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

Semiclassical Resolvent Estimates for $$L^\infty $$ Potentials on Riemannian Manifolds

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.

Second-Order Differential Operators with Non-Local Ventcel's Boundary Conditions

Different boundary conditions have been introduced for second-order differential operators and the properties of the operators on the corresponding domains have been deeply investigated since the work of Feller. The aim of this paper is to study second-order differential operators satisfying a Ventcel's type boundary condition which involves simultaneously both the endpoints of a real interval. We study different general properties and a resolvent estimate for this kind of operators.

Some lower bound estimates for resolvents of a compact operator on an infinite-dimensional Hilbert space

This paper presents some lower estimates for resolvents of a compact operator A on an infinite-dimensional Hilbert space. Usually, upper estimates for resolvents are known under suitable conditions but lower estimates are not known. These estimates enable us to calculate suitability results for approximation of a resolvent.

Uniform resolvent estimates for Schrödinger operator with an inverse-square potential

We study the uniform resolvent estimates for Schrödinger operator with a Hardy-type singular potential. Let LV=−Δ+V(x) where Δ is the usual Laplacian on Rn and V(x)=V0(θ)r−2 where r=|x|,θ=x/|x| and V0(θ)∈C1(Sn−1) is a real function such that the operator −Δθ+V0(θ)+(n−2)2/4 is a strictly positive operator on L2(Sn−1). We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator LV.

Semiclassical Estimates for Scattering on the Real Line

We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.

Exponential stability of laminated Timoshenko beams with boundary/internal controls

For a laminated Timoshenko system with slip along the interface, we study existence and boundary/interior stabilization of solutions. Our first main result, using resolvent estimates in the frequency domain, is to show that interior damping created by the interfacial slip in companion with boundary feedback controls acting on the complementary displacements (traverse and rotation angle) drives the solution to zero with exponential decay rate. We impose neither conditions on the physical parameters of the model nor on the damping coefficients. Our second main result is to investigate the asymptotic behavior of solutions for partially dissipative problems. Assuming the condition that the waves travel at the same speed, we prove the decay of solutions with exponential rate. Finally, the same condition allows us to show that dissipative effect created by the interfacial slip is strong enough to exponentially stabilizes the solutions to zero.

On Pseudospectral Bound for Non-selfadjoint Operators and Its Application to Stability of Kolmogorov Flows

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier–Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $$\nu $$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $$O(\nu ^{-\frac{1}{2}})$$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $$O(\nu ^{-1})$$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.