K. Pravda-Starov - Academia.edu (original) (raw)
Papers by K. Pravda-Starov
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prov... more We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a c... more We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R d), implying the ultra-analyticity of both the fluctuation and its Fourier transform, for any positive time.
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-c... more We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. 1 2µ) α∈N d l 2 (N d) < +∞ ⇔ f ∈ L 2 (R d), ∃t 0 > 0, e t 0 H 1/2µ f L 2 < +∞. This characterization proves that there is a regularizing effect for the solutions to the Cauchy problem (1.1) in the symmetric Gelfand-Shilov space S 1/2
Mathematische Nachrichten, 2017
We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian th... more We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.
Bulletin des Sciences Mathématiques, 2017
We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoin... more We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators.
Kinetic & Related Models, 2013
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a c... more We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R d), implying the ultra-analyticity of both the fluctuation and its Fourier transform, for any positive time.
Séminaire Laurent Schwartz — EDP et applications, 2011
Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator
Kinetic & Related Models, 2013
In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially a... more In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially as a fractional Laplacian. In the present work, we prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
Journal of Functional Analysis, 2015
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prov... more We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
Journal of Mathematical Analysis and Applications, 2015
We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L 2 spaces with respect to invar... more We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L 2 spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004
For Schrödinger operators with complex-valued potentials, we give a sufficient geometrical condit... more For Schrödinger operators with complex-valued potentials, we give a sufficient geometrical condition on potentials for the existence of pseudo-spectra. We construct some approximate semi-classical modes using a complex WKB method.
Journal of the London Mathematical Society, 2006
We study in this paper the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra ... more We study in this paper the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane which describe where the resolvent is large in norm. The study of such subsets allows to understand the stability of the spectrum under perturbations and the possible calculation of 'false eigenvalues' far from the spectrum by algorithms for eigenvalues computing. The rotated harmonic oscillator is the simplest classical non-self-adjoint quadratic Hamiltonian, which has been already studied by E.B. Davies and L.S. Boulton. In one of his work, L.S. Boulton states a conjecture about the pseudo-spectra of this operator, which describes the instabilities for the high energies. We can deduce this conjecture from a study of N. Dencker, J. Sjöstrand and M. Zworski, which gives resolvent's bounds for semi-classical pseudodifferential operator in a very general setting. In the present article, we give a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable.
Journal of Functional Analysis, 2012
We study the problem of convergence to equilibrium for evolution equations associated to general ... more We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complex-valued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and the generalized Langevin equation are given.
Journal of Differential Equations, 2014
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-c... more We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. 1 2µ) α∈N d l 2 (N d) < +∞ ⇔ f ∈ L 2 (R d), ∃t 0 > 0, e t 0 H 1/2µ f L 2 < +∞. This characterization proves that there is a regularizing effect for the solutions to the Cauchy problem (1.1) in the symmetric Gelfand-Shilov space S 1/2
Journal de Mathématiques Pures et Appliquées, 2013
The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann op... more The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.
International Mathematics Research Notices, 2010
We are interested in this paper in studying the spectral stability under small perturbations of a... more We are interested in this paper in studying the spectral stability under small perturbations of a class of non-self-adjoint semi-classical operators in dimension one. We prove that when the principal symbol's Hessian at a critical point of such an operator defines in the Weyl quantization a non-normal operator, it occurs some strong spectral instabilities under small perturbations near this critical value in a particular set where the spectrum can lie deeply inside.
Journal de Mathématiques Pures et Appliquées, 2011
We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type e... more We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prov... more We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a c... more We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R d), implying the ultra-analyticity of both the fluctuation and its Fourier transform, for any positive time.
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-c... more We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. 1 2µ) α∈N d l 2 (N d) < +∞ ⇔ f ∈ L 2 (R d), ∃t 0 > 0, e t 0 H 1/2µ f L 2 < +∞. This characterization proves that there is a regularizing effect for the solutions to the Cauchy problem (1.1) in the symmetric Gelfand-Shilov space S 1/2
Mathematische Nachrichten, 2017
We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian th... more We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.
Bulletin des Sciences Mathématiques, 2017
We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoin... more We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators.
Kinetic & Related Models, 2013
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a c... more We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R d), implying the ultra-analyticity of both the fluctuation and its Fourier transform, for any positive time.
Séminaire Laurent Schwartz — EDP et applications, 2011
Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator
Kinetic & Related Models, 2013
In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially a... more In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially as a fractional Laplacian. In the present work, we prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
Journal of Functional Analysis, 2015
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prov... more We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
Journal of Mathematical Analysis and Applications, 2015
We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L 2 spaces with respect to invar... more We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L 2 spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004
For Schrödinger operators with complex-valued potentials, we give a sufficient geometrical condit... more For Schrödinger operators with complex-valued potentials, we give a sufficient geometrical condition on potentials for the existence of pseudo-spectra. We construct some approximate semi-classical modes using a complex WKB method.
Journal of the London Mathematical Society, 2006
We study in this paper the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra ... more We study in this paper the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane which describe where the resolvent is large in norm. The study of such subsets allows to understand the stability of the spectrum under perturbations and the possible calculation of 'false eigenvalues' far from the spectrum by algorithms for eigenvalues computing. The rotated harmonic oscillator is the simplest classical non-self-adjoint quadratic Hamiltonian, which has been already studied by E.B. Davies and L.S. Boulton. In one of his work, L.S. Boulton states a conjecture about the pseudo-spectra of this operator, which describes the instabilities for the high energies. We can deduce this conjecture from a study of N. Dencker, J. Sjöstrand and M. Zworski, which gives resolvent's bounds for semi-classical pseudodifferential operator in a very general setting. In the present article, we give a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable.
Journal of Functional Analysis, 2012
We study the problem of convergence to equilibrium for evolution equations associated to general ... more We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complex-valued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and the generalized Langevin equation are given.
Journal of Differential Equations, 2014
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-c... more We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. 1 2µ) α∈N d l 2 (N d) < +∞ ⇔ f ∈ L 2 (R d), ∃t 0 > 0, e t 0 H 1/2µ f L 2 < +∞. This characterization proves that there is a regularizing effect for the solutions to the Cauchy problem (1.1) in the symmetric Gelfand-Shilov space S 1/2
Journal de Mathématiques Pures et Appliquées, 2013
The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann op... more The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.
International Mathematics Research Notices, 2010
We are interested in this paper in studying the spectral stability under small perturbations of a... more We are interested in this paper in studying the spectral stability under small perturbations of a class of non-self-adjoint semi-classical operators in dimension one. We prove that when the principal symbol's Hessian at a critical point of such an operator defines in the Weyl quantization a non-normal operator, it occurs some strong spectral instabilities under small perturbations near this critical value in a particular set where the spectrum can lie deeply inside.
Journal de Mathématiques Pures et Appliquées, 2011
We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type e... more We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.