Anisotropic vector fields: quantitative estimates and applications to the Vlasov-Poisson equation

The transport equation
$$
\partial_t u + b \cdot \nabla u=0
$$
models several physical phenomena arising in fluid dynamics and kinetic theory. The solution $u$ represents for istance the density of a collection of particles advected by the vector field $b$, which often depends on other physical quantities and $u$ itself. Nevertheless, an understanding of the linear equation is a basic step for the treatment of such nonlinear cases. When $b$ is regular the theory of characteristcs provides a link between this PDE and the ODE
$$
\dot{X}(s,x) = b(s,X(s,x)).
$$
When dealing with problems originating from mathematical physics, however, the vector field is often highly irregular, and this prevents the application of the classical theory. When $b$ has Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio shows wellposedness for the transport equation and gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. An alternative approach has been developed by Crippa-DeLellis, working at the level of the ODE and deriving a priori estimates for the flow which rely only on the Sobolev regularity and growth of $b$ (without assumptions on the divergence). They optain an upper bound for the difference between two flows, which eventually leads to uniqueness, stability and compactness (and therefore existence) of Lagrangian flows, as well as wellposdness of Lagrangian solutions to the transport equation. This estimate is derived exploiting a functional measuring a "logarithmic distance'" between two flows associated to the same vector field, namely
$$
\Phi_\delta(s) = \int \log \left( 1 + \frac{|X(s,x)-\bar X(s,x)|}{\delta}\right) \, dx \,,
$$
where $\delta>0$ is a small parameter which is optimized in the course of the argument.
Exploiting an anisotropic version of the previous functional Bohun-Bouchut-Crippa were able to recover well-posedness for the ordinary differential equation when the velocity field has anisotropic regularity (gradient given by the singular integral of an $L^{1}$ function in some directions, and the singular integral of a measure in others). In this thesis we study two applications of the last result. In the first one we prove existence of Lagrangian solutions to the Vlasov Poisson equation with point-charge. In the second one we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form
$$
b(x_1,x_2) = (b_1(x_1),b_2(x_1,x_2)) \in R^{n_1}\times R^{n_2} \,, \qquad (x_1,x_2)\in R^{n_1}\times R^{n_2}\, ,
$$
where $b_1$ has Sobolev $W^{1,p}$ regularity in $x_1$, for some $p>1$ and $b_2$ has Sobolev $W^{1,p}$ regularity in $x_2$, but only fractional Sobolev $W^{\alpha,1}$ regularity in the variable $x_1$, for some $\alpha>1/2$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field.