Regularizing a nonlinear integroparabolic Fokker--Planck equation with space-periodic solutions: existence of strong solutions (Q2773671)

Let \(Q_T=\{(\theta,\omega,t,\Omega)\in [0,2\pi]\times {\mathbb R}\times [0,T]\times [-G,G]\}\). The authors study the following boundary value problem: NEWLINE\[NEWLINE \frac{\partial\rho}{\partial t}= \frac{\partial^2\rho}{\partial \omega^2}-\omega \frac{\partial\rho}{\partial \theta} + \frac{\partial (\omega\rho)}{\partial \omega}- \Omega \frac{\partial\rho}{\partial \omega}-K_s(\theta,t) \frac{\partial\rho}{\partial \omega},\quad\rho|_{\theta=0}=\rho|_{\theta=2\pi},\;\rho|_{t=0}=\rho_0(\theta,\omega,\Omega), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE K_s(\theta,t)=k\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{0}^{2\pi}g(\Omega)\sin(\varphi-\theta) \rho(\varphi,\omega,t,\Omega) d\varphi d\omega d\Omega. NEWLINE\]NEWLINE Under some natural conditions for the initial data and the integral operator \(K_s\), it is proven that a solution \(\rho\) to this problem exists. This solution possesses the following properties: \(\rho(\theta,\omega,t,\Omega)\in W_2^{2,3,1,2}(Q_T)\) (the symbol \(W_2^{2,3,1,2}(Q_T)\) stands for the conventional anisotropic Sobolev space); \(\rho(\theta,\omega,t,\Omega)\in C^{\lambda,\lambda,1/12,1/2}(D_R)\) for every \(\lambda\in (0,1)\) and \(R>0\), where \(D_R=Q_T\cap \{\omega\in [-R,R]\}\) and \(C^{\lambda,\lambda,1/12,1/2}(D_R)\) is an anisotropic Hölder space; \(\rho(\theta,\omega,t,\Omega)\geq 0\) in \(Q_T\) and NEWLINE\[NEWLINE \int_{0}^{2\pi}\int_{-\infty}^{\infty}\rho(\theta,\omega,t,\Omega) d\omega d\theta=1,\quad t\in [0,T],\;\Omega\in [-G,G]. NEWLINE\]NEWLINE The last normalization condition as well as this problem itself arise in physics.