1. Landau equation
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Inhomogeneous Landau Equation
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
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
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.