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This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces.…
As is widely recognized in Lyapunov analysis, linearized Hamilton's equations of motion have two marginal directions for which the Lyapunov exponents vanish. Those directions are the tangent one to a Hamiltonian flow and the gradient one of…
We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…
This paper concerns with the compressible two-fluid model with algebraic pressure closure. We prove a conditional weak-strong uniqueness principle, meaning that a finite energy weak solution, with bounded densities, coincides with the…
We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the…
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity…
We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial…
The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in $d=4-\varepsilon$ dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal…
The complex potentials representing flows around a vertical plate semi-submerged in a uniform stream are derived in analytical forms by the reduction method. They are composed from the regular solution and a weak singular eigen solution.…
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the…
We study stationary solutions to the continuity equation for weakly compressible flows. These describe non-equilibrium steady states of weakly dissipative dynamical systems. Compressibility is a singular perturbation that changes the steady…
The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying…
We consider the following stochastic partial differential equation, \begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=\psi, \end{align*} associated with a stochastic flow $\{X(t,x)\}$, for $t \geq 0$, $x \in \mathbb{R}^d$, as in…
In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through…
We explain the construction of some solutions of the Stokes system with a given set of singular points, in the sense of Caffarelli, Kohn and Nirenberg. By means of a partial regularity theorem (proved elsewhere), it turns out that we are…
In this paper, we consider a two-phase flow model consisting of the compressible Navier-Stokes systems with degenerate viscosity coupled with the compressible Navier-Stokes systems with constant viscosities via a drag force, which can be…
We present a convergence analysis of a finite volume (FV) scheme for the multicomponent compressible Euler system in the framework of dissipative weak (DW) solutions. DW solutions were introduced as a generalized solution framework in…
An invariant measure for a flow is, of course, an invariant measure for any of its time-t maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the time-one map to…
For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal…
This paper is about well-posedness and realizability of the kinetic equation for gas-particle flows and its relationship to the Generalized Langevin Model (GLM) PDF equation. Previous analyses claim that this kinetic equation is ill-posed,…