AMS: 35B45, 42B20, 60J75, 47G30
DOI: 10.21538/0134-4889-2025-31-1-210-227
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 equat\i on 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
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Received September 5, 2024
Revised November 15, 2024
Accepted November 18, 2024
Veli Shakhmurov, Dr. Phys.-Math. Sci., Prof., Antalya Bilim University Department of Industrial Engineering, Dosemealti, 07190 Antalya, Turkey, e-mail: veli.sahmurov@antalya.edu.tr; Azerbaijan State Economic University, Center of Analytical-Information Resource, 194 M. Mukhtarov AZ1001 Baku, Azerbaijan, veli.sahmurov@gmail.com; Physics and Technical Sciences, Western Caspian University, Baku, AZ1001 Azerbaijan.
Cite this article as: Veli Shakhmurov. Kinetic maximal $L^p$ -regularity for nonlocal Kolmogorov equation and application. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2025, vol. 31, no. 1, pp. 210–227.
Русский
Вели Шахмуров. Кинетическая максимальная $L^p$-регулярность для нелокального уравнения Колмогорова и ее применение
Мы изучаем линейные и нелинейные уравнения Колмогорова с переменными коэффициентами. Уравнения включают абстрактный оператор $A=A\left(x\right) $ в банаховом пространстве типа Фурье $E$ и члены свертки. Здесь в терминах $E$-значных пространств Соболева выводится кинетическая максимальная $L^{p}$-регулярность для линейного уравнения. Более того, мы показываем, что решение $u$ также регулярно во времени и пространстве переменных, если предполагается, что $u$ имеет определенную регулярность по скорости. Наконец, кинетическая максимальная $L^{p}$-регулярность для линейного уравнения может быть использована для получения локального существования и единственности решений квазилинейного нелокального кинетического уравнения колмогоровского типа.
Ключевые слова: Кинетическая максимальная регулярность, уравнение Колмогорова, диссипативные операторы, анизотропные пространства Соболева, оптимальные $L^{p}$-оценки, мгновенное сглаживание
