Nonlinear evolution and transport equations for measures (Q845483)

The authors prove the existence of weak solution to the Cauchy problem for the following transport equation: \[ \partial_t \mu_t + \text{div}_x (b(\mu,x,t)\,\mu_t)=0, \quad (x,t)\in \mathbb R^d\times [0,1], \] where \(\mu=(\mu_t)_{t\in [0,1]}\) and \(\nu\) are probability measures. The main contribution consists in proving the existence of solution only under growth conditions on the coefficient \(b\): \(\langle b(\mu,x,t),x\rangle \leq c(1+|x|^2)\) and \(|b(\mu,x,t)|\leq c (1+|x|^{2k})\) for some \(k\in \mathbb N\) and all \(x\in \mathbb R^d\) and \(\mu\). In this sense, existing results in literature, e.g. for the linear case, require some regularity of \(b\) with respect to the space variable [see \textit{L. Ambrosio}, Invent. Math. 158, No. 2, 227--260 (2004; Zbl 1075.35087)]. The proof of the main result is based on the method of vanishing viscosity [see e.g. \textit{R. J. Di Perna} and \textit{P. L. Lions}, Invent. Math. 98, 511--548 (1989; Zbl 0696.34049)] and Schauder's fixed point theorem on a certain space of absolutely continuous probability measures.