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Adaptive Finite Element Methods For Parabolic Problems. VI. Analytic Semigroups
 SIAM J. Numer. Anal
, 1998
"... . We continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups. We prove optimal a priori and a posteriori error estimates for the discontinuous Galerkin method showing, in particular, that analytic semigroups allow longtime integration without ..."
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Cited by 123 (3 self)
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. We continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups. We prove optimal a priori and a posteriori error estimates for the discontinuous Galerkin method showing, in particular, that analytic semigroups allow longtime integration without error accumulation. 1. Introduction This paper is a continuation of the series of papers [1], [2], [3], [4], [5] on adaptive finite element methods for parabolic problems. The method considered is the discontinuous Galerkin method (the dGmethod) based on a spacetime finite element discretization with piecewise polynomial basis functions that are continuous in space and discontinuous in time. In [1], [2], [3], [4], [5] we proved optimal a priori and a posteriori error estimates for the dGmethod for parabolic problems, typically of the form: find u : [0; 1) ! H such that u(t) + Au(t) = f(t); t ? 0; u(0) = u 0 ; (1.1) where H = L 2 (\Omega\Gamma with\Omega a bounded domain in R n , Av = ...
Pseudospectra of linear operators
 SIAM Rev
, 1997
"... Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A ..."
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Cited by 113 (8 self)
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Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
Transfer functions of regular linear systems Part III: Inversions And Duality
 Trans. Amer. Math. Soc
, 2000
"... We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing ..."
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Cited by 78 (13 self)
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We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing both of the inversions mentioned before. A wellposed linear system is timeinvertible if and only if its operator semigroup extends to a group. The system is flowinvertible if and only if its inputoutput map has a bounded inverse on some (hence, on every) finite time interval [0; ] ( > 0). This is true if and only if the transfer function of has a uniformly bounded inverse on some right halfplane. The system is timeflowinvertible if and only if on some (hence, on every) finite time interval [0; ], the combined operator from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conservative, since then is unitary. Timeowinversion can sometimes, but not always, be reduced to a combination of time and flowinversion. We derive a surprising necessary and sucient condition for to be timeflowinvertible: its system operator must have a uniformly bounded inverse on some left halfplane.
Infinitedimensional linear systems with unbounded control and observation: A functional analytic approach
 Transactions of the American Mathematical Society
, 1987
"... ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and ..."
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Cited by 59 (1 self)
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ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. 1. Introduction. For
Stability of Travelling Waves
, 2002
"... An overview of various aspects related to the spectral and nonlinear stability of travellingwave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm p ..."
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Cited by 46 (7 self)
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An overview of various aspects related to the spectral and nonlinear stability of travellingwave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of wellseparated multibump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.
The diffusion limit of transport equations derived from velocity jump processes
 Siam J. Appl. Math
, 2000
"... Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations c ..."
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Cited by 41 (14 self)
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Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropicdiffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak–Keller–Segel–Alt model for chemotaxis.
Kinetic Models for Chemotaxis and their DriftDiffusion Limits
, 2003
"... Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is pro ..."
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Cited by 38 (11 self)
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Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is proven.
Stability of largeamplitude shock waves of compressible NavierStokes equations
, 2003
"... We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be ..."
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Cited by 38 (24 self)
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We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multidimensions by a codimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently smallamplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently largeamplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory.
Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin
 Dyn. Syst
"... Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥ ..."
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Cited by 36 (0 self)
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Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥3thegrowth condition f(u)  ≤c0(u  γ +1), where1≤γ≤ n. No Lipschitz condition on f n−2 is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser’s property, which implies the connectedness of the attractor. In the case n ≥ 3andγ> n the existence of a global attractor is proved under n−2 the (unproved) assumption that every weak solution satisfies the energy equation. Dedicated to M.I. Vishik on the occasion of his 80 th birthday