arXiv subjects

Gayrat Toshpulatov

Publications and source records attributed to Gayrat Toshpulatov.

10 recordsLinked to original sources

arXiv2026

Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation

This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor $f(1-f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint $0\leq f\leq 1$. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation $(BV),$ allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.

math.AP↗

arXiv2026

Long time behavior of discrete velocity kinetic equations

We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the $L^2$ space. Our result holds for a wide class of interaction rates including the Goldstein-Taylor and Carleman equations, and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of suitable Lyapunov functionals by modifying Boltzmann's entropy.

math.AP↗

arXiv2026

Long time behavior of Fokker-Planck equations for bosons and fermions

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor $f(1\pm f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the inclusion principle of bosons and the exclusion principle of fermions. Assuming that global solutions exist, we prove exponential decay of the solutions to the global equilibrium in a weighted $L^2$-space without a close-to-equilibrium assumption. Our analysis is in the spirit of an $L^2$-hypocoercivity method. Our main Lyapunov functional is constructed from a logarithmic relative entropy and the (nonlinear) projection of the solution to the manifold of local-in-$x$ equilibria.

math.AP↗

arXiv2026

Convergence to global equilibrium for the semiconductor Boltzmann equation

The Boltzmann equation describing the transport of electrons in semiconductor devices with an external electrostatic potential is considered when the spatial variable is in a torus and the wave vector is in the Brillouin zone. We prove the exponential time decay of solutions towards the global equilibrium in a weighted $L^2$ space. Our result holds for wide classes of energy functions of electrons and external electrostatic potentials, and the estimates on the rate of convergence are explicit and constructive. We remove the close-to-equilibrium assumption on the initial datum and the parabolic band approximation assumption that were required in previous works. The technique is based on the construction of a suitable highly non-linear Lyapunov functional by modifying the relative entropy. The analysis benefits from uniform bounds for the solution in terms of the global equilibrium.

math.AP↗

arXiv2026

Sharp estimates for the Fourier transform of surface-carried measures and maximal operators associated with hypersurfaces in $\mathbb{R}^4$ with vanishing Gaussian curvature

In this paper, we study problems related to harmonic analysis on hypersurfaces in $\mathbb{R}^4 $ with zero Gaussian curvature and given as graphs of polynomial functions. We derive sharp uniform estimates with respect to the direction of frequencies for the Fourier transform of measures supported on such hypersurfaces. Additionally, we study the $L^p$-boundedness problem of maximal operators associated with hypersurfaces. We determine the exact value of the boundedness exponent in terms of the heights of these hypersurfaces.

math.CA↗

arXiv2025

Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation

We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted $L^2$ and Sobolov spaces. Our result holds for a wide class of external potentials and the estimates on the rate of convergence are explicit and constructive. Moreover, we prove that the associated semigroup of the equation has hypoelliptic regularizing properties and we obtain explicit rates on this regularization. The technique is based on the construction of suitable Lyapunov functionals.

math.AP↗

arXiv2024

Hypocoercivity for the non-linear semiconductor Boltzmann equation

A kinetic model for semiconductor devices is considered on a flat torus. We prove exponential decay to equilibrium for this non-linear kinetic model by hypocoercivity estimates. This seems to be the first hypocoercivity result for this nonlinear kinetic equation for semiconductor devices without smallness assumptions. The analysis benefits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

math.AP↗

arXiv2024

Well-posedness and trend to equilibrium for the Vlasov-Poisson-Fokker-Planck system with a confining potential

We study well-posedness and long time behavior of the nonlinear Vlasov-Poisson- Fokker-Planck system with an external confining potential. The system describes the time evolution of particles (e.g.$\,\,$in a plasma) undergoing diffusion, friction and Coulomb interaction. We prove existence and uniqueness of mild solutions in a weighted Sobolev space. Moreover, we prove that the solutions converge to the global equilibrium exponentially. Our results hold for a wide class of external potentials and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of Lyapunov functionals, new short and long time estimates for the linearized system, and fixed point arguments.

math.AP↗

arXiv2024

Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted $H^1$-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted $L^2$-distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order $\mathcal O\big( (1+t)e^{-tν/2}\big)$, with $ν$ the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted $L^2$-space to a weighted $H^1$-space).

math.AP↗

arXiv2022

Strong $L^2 H^2$ convergence of the JKO scheme for the Fokker-Planck equation

Following a celebrated paper by Jordan, Kinderleherer and Otto it is possible to discretize in time the Fokker-Planck equation $\partial_t\varrho=Δ\varrho+\nabla\cdot(ρ\nabla V)$ by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in $L^1$ in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to $W^{1,p}$ for an exponent $p$ larger than the dimension), that the convergence is actually strong in $L^2_tH^2_x$, hence strongly improving the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.

math.OC↗