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6. Analysis of nonlocal equations

    1. General $L^p \to L^\infty$ estimate


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      Problem 6.1.

      Prove $L^p \to L^\infty$ estimate for $$\mathscr{L}u = f$$ for $f \in L^p$ and $\mathscr{L}$ a non-divergence, nonlocal form.

        • Continuity of solutions for multi-dimensional $p$-Laplacian


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          Problem 6.2.

          Prove continuity of solutions for the system case of the fraction $p$- Laplacian.

            • Schauder estimate for general transport term


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              Problem 6.4.

              Obtain Schauder estimates for a general transport term, $$(\partial_t + b(v)\cdot \nabla)f - Tr(AD^2_v\cdot).$$

              Additionally, kinetic formulation of systems of conservation laws yield transport operators that look like $$(\partial_t + b(t,x,v)\cdot \nabla_x ).$$ Can one obtain Schauder estimates for this further generalized transport?

                • Uniqueness for porous medium in $d > 1$

                      The porous medium equation is given by $$\partial_t u = \nabla_x \cdot(u\nabla_x\Delta^{-s}_x u)$$ and lacks a comparison principle in $d > 1$.

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                  Problem 6.6.

                  Prove uniqueness of solutions for the porous medium equation in $d > 1$ with not too smooth initial data. Is the Wasserstein distance contractive?

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