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8. Linear kinetic equations

    1. Regularity up to boundary for kinetic equations

          Consider a boundary in $\R^d_x$, that is in $x$. Since transport is first order in $x$, one can only define boundary conditions when the velocity points in a particular direction (e.g. normal to the boundary).

      In the case of the homogeneous Fokker-Planck equation on the half-space (constant coefficients) the regularity of solutions up to the boundary is $C_v^{1/2}C_x^{1/6}$ [Henderson, Lucertini, Wang 2024].

      With no assumption on the regularity of coefficients, the regularity of solutions up to the boundary is $C^\alpha$ for some $\alpha$.

      Add remark

      Problem 8.1.

      [Christopher Henderson] What is the relationship between the regularity of the coefficients and the boundary regularity? Is $C_x^{1/6}C_v^{1/2}$ regularity sharp in the case of constant coefficients?

        • Regularity of solutions of non-divergence form of a linear kinetic equation


          Add remark

          Problem 8.2.

          Study the regularity of solutions a kinetic equation $$\partial_t f + v\cdot \nabla_x f = \mathscr{L}f$$ where $\mathscr{L}$ is linear and of non-divergence form. In the local case, $\mathscr{L} = Tr(AD^2_v\cdot)$.

          If $\mathscr{L} = (\Delta)^s$, is there a monotone quantity?

            • Inverse problem for interacting particle system

                  Consider the Vlasov-Maxwell equations $$\begin{cases} (\partial_t + v\cdot \nabla_x)f = -(E + v\times B)\cdot \nabla_v f\\ -\Delta u = \rho\\ \nabla u = E\end{cases}.$$

              Add remark

              Problem 8.3.

              Assume that one can tune any macroscopic quantity (i.e. $\rho(t,x)$, $E$, $B$ etc.) What is necessary to recover the initial distribution $f_0$ for $d \geq 2$? Is this possible?

              There is perhaps a natural extension of this problem to the wave kinetic equation. What would the additional quantity to control be in this case?

                  Cite this as: AimPL: Integro-differential equations in many-particle interacting systems, available at http://aimpl.org/manyparticle.