Kinetic maximal $L^{p}$-regularity for nonlocal Kolmogorov equation and application

V. B. Shakhmurov<sup>abc

a</sup> Antalya Bilim University
^b Azerbaijan State Economic University
^c Western Caspian University

Abstract: We study the linear and nonlinear variable coefficients Kolmogorov equations. The equations include the abstract operator $A=A\left(x\right) $ in a Fourier type Banach space $E$ and convolution terms. Here, the kinetic maximal $L^{p}$-regularity for the linear equatıon is derived in terms of $ E$-valued Sobolev spaces. Moreover, we show that the solution $u$ is also regular in time and space variables when $u$ is assumed to have a certain amount of regularity in velocity. Finally, the kinetic maximal $L^{p}$ -regularity for the linear equation can be used to obtain local existence and uniqueness of solutions to a quasilinear nonlocal Kolmogorov type kinetic equation.

Keywords: Kinetic maximal regularity, Kolmogorov equation, dissipative operators, anisotropic Sobolev spaces, optimal $L^{p}$ -estimates, instantaneous smoothing.

Received: 05.09.2024
Revised: 15.11.2024
Accepted: 18.11.2024

Language: English

DOI: 10.21538/0134-4889-2025-31-1-210-227

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