Related papers: Interpolation inequalities, nonlinear flows, bound…
This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical R\'enyi…
We consider the numerical integration of conservation laws endowed with an entropy inequality and we study the residual of the scheme on this inequality, which represents the numerical entropy production. This idea has been introduced and…
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also…
We present a novel technique for imposing non-linear entropy conservative and entropy stable wall boundary conditions for the resistive magnetohydrodynamic equations in the presence of an adiabatic wall or a wall with a prescribed heat…
Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice…
The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It…
In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use…
We develop a practical and novel method for inference on intersection bounds, namely bounds defined by either the infimum or supremum of a parametric or nonparametric function, or equivalently, the value of a linear programming problem with…
The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid-structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an…
We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and…
For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite element method (FEM). In solving the optimization…
We present a new hydrodynamic model for incompressible binary fluids that is thermodynamically consistent and non-isothermal. This model follows the generalized Onsager principle and Boussinesq approximation and preserves the volume of each…
A Massey-like inequality is any useful lower bound on guessing entropy in terms of the computationally scalable Shannon entropy. The asymptotically optimal Massey-like inequality is determined and further refined for finite-support…
In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\geq r$, $\delta\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{\delta…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^\gamma\mathrm{div}\left(|x|^{-\beta}\nabla u^m\right)$ posed on…
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the…
We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the…
This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents.…
Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of…