相关论文: Rectifiability of entropy productions for weak sol…
The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colon\Omega \subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(\Omega),\quad |m|=1\text{ a.e.…
A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colon\Omega\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(\Omega)$. It is known that, if $m$ has some low…
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose $\Gamma$-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in…
We study the fine properties of a class of weak solutions $u$ of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in (Rivi\`ere T, Serfaty S. Limiting domain wall energy for a problem related to…
Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a…
Characterizing rectifiability of Radon measures in Euclidean space has led to fundamental contributions to geometric measure theory. Conditions involving existence of principal values of certain singular integrals…
This paper provides tight bounds on the R\'enyi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the R\'enyi…
Weakly almost i.i.d. quantum sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We establish…
Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically…
Consider the set of all sequences of $n$ outcomes, each taking one of $m$ values, that satisfy a number of linear constraints. If $m$ is fixed while $n$ increases, most sequences that satisfy the constraints result in frequency vectors…
A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on…
We establish a weak-strong uniqueness principle for solutions to entropy-dissipating reaction-diffusion equations: As long as a strong solution to the reaction-diffusion equation exists, any weak solution and even any renormalized solution…
Motivated by the desire to improve our understanding of the Weak Gravity Conjecture, we compute the one-loop correction of charged particles to the geometry and entropy of extremal black holes in 4d. We find that fermion loops provide…
Predictive inference requires balancing statistical accuracy against informational complexity, yet the choice of complexity measure is usually imposed rather than derived. We treat econometric objects as predictive rules, mappings from…
Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon \left|\nabla m\right|^2 +…
Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, an approximate solution sequence is constructed, and then a priori estimates are…
Many partially-successful attempts have been made to find the most natural discrete-variable version of Shannon's entropy power inequality (EPI). We develop an axiomatic framework from which we deduce the natural form of a discrete-variable…
Relative entropy is the standard measure of distinguishability in classical and quantum information theory. In the classical case, its loss under channels admits an exact chain rule, while in the quantum case only asymptotic, regularized…
We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection…
The weak law of large numbers implies that, under mild assumptions on the source, the Renyi entropy per produced symbol converges (in probability) towards the Shannon entropy rate. This paper quantifies the speed of this convergence for…