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The flow associated to weakly differentiable vector fields. (English) Zbl 1178.35134
Tesi. Scuola Normale Superiore Pisa (Nuova Serie) 12. Milano: Springer; Pisa: Scuola Normale Superiore (Dissertation) (ISBN 978-88-7642-340-6/pbk). xv, 167 p. (2009).
This paper is a study of the ordinary differential equations $\dot \gamma (t) = b(t, \gamma(t)), \quad \gamma(0) = x, \quad \gamma: [0,T] \to \mathbb R^d, \eqno(1)$ and the corresponding transport equations $\partial_t u(t,x) + b(t,x) \cdot \nabla_x u(t,x) = 0, \quad u(0,x) = {\bar u}(x), \quad u: [0,T] \times \mathbb R^d \to \mathbb R^1, \eqno(2)$ where the vector field $$b: [0,T] \times \mathbb R^d \to \mathbb R^d$$ satisfies some regularity assumptions weaker that the classical (continuity and the Lipschtz condition by $$x$$). For instance, the author introduces the notion of weak solution of (2) in the standard sense assuming that $$b(t,x), {\bar u}(x)$$ are locally summable, and $$\text{div}_x\,b(t,x)$$ is locally summable (in the sense of distributions). Under these suppositions he defines so-called renormalized solution of (2): a weak solution $$u \in L^{\infty}$$ is a renormalized one if for every function $$\beta \in C^1(\mathbb R^1;\mathbb R^1)$$ the equality $$\partial_t (\beta(u)) + b(t,x) \cdot \nabla_x (\beta(u)) = 0$$ holds (in the sense of distributions). The author proves that the renormalization property of the vector field $$b$$ (this means that every bounded solution of (2) with given $$b$$ is renormalized one) implies well-posedness of the equation (2); he establishes the renormalization property for some classes of vector fields and consequently well-posedness of corresponding transport equations. The author also studies the relations between the notions of renormalization, strong continuity of the solution with respect to the time, forward and backward uniqueness in the Cauchy problem and approximation of the solution with smooth maps.
In the second part of the paper the author describes the connection between ODE (1) and PDE (2) out of the classical case. He considers suitable measures in the space of continuous maps and establishes the equivalence between pointwise uniqueness for the ODE and uniqueness of positive measure-valued solutions of the continuity equation, exploiting the superposition principle for solutions of the PDE. Then he introduces the theory of regular Lagrangian flows developed by L. Ambrosio, and shows how the uniqueness of bounded solutions of the continuity equation implies existence and uniqueness of the regular Lagrangian flow. At last, he obtains some quantitative estimates for the vector fields of the class $$W^{1,p}$$, $$p > 1$$, which allow to recover the results of existence, uniqueness and stability of regular Lagrangian flows obtained before, and to get some consequences regarding compactness, quantitative regularity and quantitative stability of regular Lagrangian flows.

##### MSC:
 35F10 Initial value problems for linear first-order PDEs 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34A36 Discontinuous ordinary differential equations 35D30 Weak solutions to PDEs