Related papers: Non-uniqueness of integral curves for autonomous H…

We prove that for bounded, divergence-free vector fields b in L^1_{loc}((0,1];BV(\T^d;\R^d)), there exists a unique incompressible measure on integral curves of b. We recall the vector field constructed by Depauw in [Depauw, C. R. Math.…

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L^1_{loc}((0, T ]; BV (\mathbb{T}^d;\mathbb{R}^d))\cap L^2((0, T )…

For linear scalar field theories, I characterize those classical Hamiltonian vector fields which have self-adjoint operators as their quantum counterparts. As an application, it is shown that for a scalar field in curved space-time (in a…

In the article we establish the global well-posedness in W^{1,(6,2)}(R \times R+) of the integro-differential equation containing the cube of the one dimensional Laplacian and the transport term. Our proof relies on a fixed point technique.…

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to…

We construct infinitely many incompressible Sobolev vector fields $u \in C_t W^{1,\tilde p}_x$ on the periodic domain $\mathbb{T}^d$ for which uniqueness of solutions to the transport equation fails in the class of densities $\rho \in C_t…

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in $\rho \in L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u \in…

A uniqueness theorem is established for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every…

We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak…

We reproduce the ${\cal N}=4$ supersymmetric mechanics on curved spaces, constructed earlier within the Hamiltonian formalism, using the superfield methods. We show that for any such mechanics, given by the metric and the third order…

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…

We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is…

We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where…

These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the…

Let $(X,d,m)$ be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided $(X,d,m)$ satisfies a new…

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic…

A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is…

Consider a Hamiltonian diffeomorphism $g$ on a surface. We describe several scenarios where a curve $L$ and its image $g(L)$ provide a simple evidence that $g$ is not autonomous.

Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form $-\Delta u(x) + \frac{1}{\gamma} |D{u}(x)|^\gamma = f(x) - \lambda$, with $f$ a coercive function and $\lambda$ a constant, in the subquadratic…

We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L^1 (0,T; W^{1,p}(\mathbb{T}^d)) \cap L^\infty ((0,T) \times \mathbb{T}^d)]^d$ with $\operatorname{div}(u) ^- \in L^1 (0,T;…