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Phase separation routinely occurs in both living and synthetic systems. These phases are often complex and distinguished by features including crystallinity, nematic order, and a host of other nonconserved order parameters. For systems at…
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…
This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data…
The GENERIC structure allows for a unified treatment of different discrete models of hydrodynamics. We first propose a finite volume Lagrangian discretization of the continuum equations of hydrodynamics through the Voronoi tessellation. We…
Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD…
Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof.…
We first give an abstract framework to show the uniqueness of Ground State Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all the existing results in the literature only addressed particular cases. Moreover, our…
In the previous companion paper, we proposed a subclass of wavefunctions to describe macroscopic solids that resolved and extended the theory quantum measurement and gave a more specific treatment of quasiparticles. Here we extend these…
The BBGKY hierarchy of equations for a particle interacting with an ideal gas is investigated. Principal properties of its solutions are disclosed, as exact identities which connect probability distribution of path of the particle, its…
Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional…
Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical…
We, for the first time, report a first-principle proof of the equations of state used in the hydrodynamic theory for integrable systems, termed generalized hydrodynamics (GHD). The proof makes full use of the graph theoretic approach to…
The paper deals with the existence and almost periodic homogenization of some model of generalized Navier-Stokes equations. We first establish an existence result for non-stationary Ladyzhenskaya equations with a given non constant density.…
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing.…
In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley…
Multicomponent phase separation is a routine occurrence in both living and synthetic systems. Thermodynamics provides a straightforward path to determine the phase boundaries that characterize these transitions for systems at equilibrium.…
Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to…
We reformulate and generalize the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed point…
Universal thermodynamics for FRW model of the Universe bounded by apparent/event horizon has been considered for massive gravity theory. Assuming Hawking temperature and using the unified first law of thermodynamics on the horizon, modified…
We discuss and illustrate through numerical examples the relations between generating functionals, thermodynamic consistency (in particular the virial-free energy one), and uniqueness of the solution, in the integral equation theory of…