and uniqueness in Section $4$. Throughout the paper the value $C > 0$ will denote a generic constant that may change from line to line. When necessary, we will specifically identify the quantities upon which $C$ may depend. Regarding norms, we will abuse notation and allow the reader to differentiate certain norms via context. For instance, $\Vert f(t) \Vert_\infty = \mathop{\esssup}\limits_{x \in \bfR,v \in \bfR^2} \vert f(t,x,v) \vert$, whereas $\Vert B(t) \Vert_\infty = \mathop{\esssup}\limits_{x \in \bfR} \vert B(t,x) \vert$, with analogous statements for ${\ensuremath{\left\| \cdot \right\|}}_2$ and ${\ensuremath{\left\langle \cdot, \cdot \right\rangle}}$ which denote the $L^2$ norm and inner product, respectively. Finally, for derivative estimates we will use the notation $$\Vert v_0^\gamma \partial_x^j {\nabla}_v^k f(t) \Vert_2 = \sum_{\vert \alpha \vert = k} \Vert v_0^\gamma \partial_x^j \partial_v^\alpha f(t) \Vert_2$$ for $\gamma \in \bfR$, $j,k = 0,1,2,...$, and a multi-index $\alpha=(\alpha_1,\alpha_2)$ where we denote ${\partial}_{v_1}^{\alpha_1}{\partial}_{v_2}^{\alpha_2}$ by ${\partial}_v^\alpha$.

 *A priori* estimates
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Let $T > 0$ be given so that we may estimate on the bounded time interval $(0,T)$ when necessary. To begin, we will first prove a result that will allow us to estimate the particle density and its moments. When studying collisionless kinetic equations, one often wishes to integrate along the Vlasov 