Sensitivity equations for measure-valued solutions to transport equations

Abstract

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R}^d)$: $ \partial_t\mu_t + \partial_x(v(x) \mu_t) = 0. $ We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $ \partial _t\mu^h_t + \partial_x(v^h(x)\mu^h_t) = 0. $ We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $\partial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R}^d)$ endowed with the dual norm $(C^{1, \alpha}(\mathbb{R}^d))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v = v[\mu_t](x)$.

Keywords

• transport equations
• space of radon measures
• differentiability of solutions
• very weak solutions