The following comes from Pages 37-41 of the following paper:


Let $\mathbb{T}^d$ be a $d$-dimensional torus and let $C^{n+ \alpha}$ be the $n$th order $\alpha$-Holder space defined in the usual way. We define functions $V \in C([0,T], (C^{n+ 1+\alpha}(\mathbb{T}^d))^d) $ and $c \in L^{\infty} \big([0,T], \big( C^{n+ \alpha}(\mathbb{T}^d) \big)' \big)$. Let $\rho$ be a measure on $\mathbb{T}^d$. We consider, in the distribution sense, the PDE given by $$ \partial_t \rho + \Delta \rho - \text{div} (\rho V)- \text{div} (c) = 0, \quad \quad \text{ on } [0,T],$$ with initial condition $\rho(0)=0$. Lemma 3.5 of the paper states that there exists a unique solution which satisfies $$ \sup_{t \in [0,T]} \| \rho (t) \|_{( C^{n+ 1+\alpha}(\mathbb{T}^d) )'} \leq C \sup_{t \in [0,T]} \| c (t) \|_{( C^{n+ \alpha}(\mathbb{T}^d) )'} < \infty. $$

My question is whether or not this holds for $\mathbb{R}^d$ instead of $\mathbb{T}^d$. I am not able to check in the paper as most of the technical details are hidden. Thanks.


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