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1. Landau equation

    1. Inhomogeneous Landau Equation


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      Problem 1.1.

      [Jacob Bedrossian (University of Maryland, College Park) \& Maria Pia Gualdani (George Washington University)] Let us consider the inhomogeneous Landau equation \partial_f+v\cdot\nabla_xf=Q(f,f), where f=f(x,v,t) and Q(f,f)=\text{div}_v\int_{\mathbb{R}^3}|v-w|^{-1}\Pi(v-w)(f(w)\nabla_vf(v)-f(v)\nabla_wf(w))\,dw,
      with \Pi(z)=\frac{1}{8\pi}(id-\frac{z\otimes z}{|z|^2}). Is it possible to derive Q from particles? This should be false in some regimes of relevance for plasma physicists. Moreover it could be very interesting to look at the weakly collisional limit for \partial_f+v\cdot\nabla_xf=\epsilon Q(f,f) as \epsilon\to0 near the equilibrium.

        • Homogeneous Landau equation


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          Problem 1.2.

          [Maria Pia Gualdani (George Washington University)] Let us now look at the homogeneous Landau equation, i.e. \partial_tf=Q(f,f), where Q is as the previous problem. Can we prove that f\in L_t^{\infty}L_v^p, for p>1? Another interesting problem could be proving (or disproving) certain type of blow-up (via a classification of them).

            • Isotropic Landau equation


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              Problem 1.3.

              [Maria Pia Gualdani (George Washington University)] What is the long-time decay of the solution toward the steady state f_{\infty}=0?

                  Cite this as: AimPL: Nonlocal differential equations in collective behavior, available at http://aimpl.org/nonlocalde.