527 research outputs found
Global classical solutions and large-time behavior of the two-phase fluid model
We study the global existence of a unique strong solution and its large-time
behavior of a two-phase fluid system consisting of the compressible isothermal
Euler equations coupled with compressible isentropic Navier-Stokes equations
through a drag forcing term. The coupled system can be derived as the
hydrodynamic limit of the Vlasov-Fokker-Planck/isentropic Navier-Stokes
equations with strong local alignment forces. When the initial data is
sufficiently small and regular, we establish the unique existence of the global
-solutions in a perturbation framework. We also provide the large-time
behavior of classical solutions showing the alignment between two fluid
velocities exponentially fast as time evolves. For this, we construct a
Lyapunov function measuring the fluctuations of momentum and mass from its
averaged quantities
Finite-time blow-up phenomena of Vlasov/Navier-Stokes equations and related systems
This paper deals with the finite-time blow-up phenomena of classical
solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the
initial configurations. We show that a solution to the coupled kinetic-fluid
system may be initially smooth, however, it can become singular in a finite
period of time. We provide a simple idea of showing the finite time blow up of
classical solutions to the coupled system which has not been studied so far. We
also obtain analogous results for related systems, such as isentropic
compressible Navier-Stokes equations, two-phase fluid equations consisting of
pressureless Euler equations and Navier-Stokes equations, and thick sprays
model
Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces
In this paper, we are concerned with the global well-posedness and
time-asymptotic decay of the Vlasov-Fokker-Planck equation with local alignment
forces. The equation can be formally derived from an agent-based model for
self-organized dynamics which is called Motsch-Tadmor model with noises. We
present the global existence and uniqueness of classical solutions to the
equation around the global Maxwellian in the whole space. For the large-time
behavior, we show the algebraic decay rate of solutions towards the equilibrium
under suitable assumptions on the initial data. We also remark that the rate of
convergence is exponential when the spatial domain is periodic. The main
methods used in this paper are the classical energy estimates combined with
hyperbolic-parabolic dissipation arguments
A sharp error analysis for the discontinuous Galerkin method of optimal control problems
In this paper, we are concerned with a nonlinear optimal control problem of
ordinary differential equations. We consider a discretization of the problem
with the discontinuous Galerkin method with arbitrary order . Under suitable regularity assumptions on the cost
functional and solutions of the state equations, we provide sharp estimates for
the error of the approximate solutions. Numerical experiments are presented
supporting the theoretical results
Cucker-Smale flocking particles with multiplicative noises: stochastic mean-field limit and phase transition
In this paper, we consider the Cucker-Smale flocking particles which are
subject to the same velocity-dependent noise, which exhibits a phase change
phenomenon occurs bringing the system from a "non flocking" to a "flocking"
state as the strength of noises decreases. We rigorously show the stochastic
mean-field limit from the many-particle Cucker-Smale system with multiplicative
noises to the Vlasov-type stochastic partial differential equation as the
number of particles goes to infinity. More precisely, we provide a quantitative
error estimate between solutions to the stochastic particle system and
measure-valued solutions to the expected limiting stochastic partial
differential equation by using the Wasserstein distance. For the limiting
equation, we construct global-in-time measure-valued solutions and study the
stability and large-time behavior showing the convergence of velocities to
their mean exponentially fast almost surely
A hydrodynamic model for synchronization phenomena
We present a new hydrodynamic model for synchronization phenomena which is a
type of pressureless Euler system with nonlocal interaction forces. This system
can be formally derived from the Kuramoto model with inertia, which is a
classical model of interacting phase oscillators widely used to investigate
synchronization phenomena, through a kinetic description under the mono-kinetic
closure assumption. For the proposed system, we first establish local-in-time
existence and uniqueness of classical solutions. For the case of identical
natural frequencies, we provide synchronization estimates under suitable
assumptions on the initial configurations. We also analyze critical thresholds
leading to finite-time blow-up or global-in-time existence of classical
solutions. In particular, our proposed model exhibits the finite-time blow-up
phenomenon, which is not observed in the classical Kuramoto models, even with a
smooth distribution function for natural frequencies. Finally, we numerically
investigate synchronization, finite-time blow-up, phase transitions, and
hysteresis phenomena.Comment: 40 pages, 37 figure
Collective behavior models with vision geometrical constraints: truncated noises and propagation of chaos
We consider large systems of stochastic interacting particles through
discontinuous kernels which has vision geometrical constrains. We rigorously
derive a Vlasov-Fokker-Planck type of kinetic mean-field equation from the
corresponding stochastic integral inclusion system. More specifically, we
construct a global-in-time weak solution to the stochastic integral inclusion
system and derive the kinetic equation with the discontinuous kernels and the
inhomogeneous noise strength by employing the 1-Wasserstein distance
Hydrodynamic Cucker-Smale model with normalized communication weights and time delay
We study a hydrodynamic Cucker-Smale-type model with time delay in
communication and information processing, in which agents interact with each
other through normalized communication weights. The model consists of a
pressureless Euler system with time delayed non-local alignment forces. We
resort to its Lagrangian formulation and prove the existence of its global in
time classical solutions. Moreover, we derive a sufficient condition for the
asymptotic flocking behavior of the solutions. Finally, we show the presence of
a critical phenomenon for the Eulerian system posed in the spatially
one-dimensional setting
Global existence of weak solutions for Navier-Stokes-BGK system
In this paper, we study the global well-posedness of a coupled system of
kinetic and fluid equations. More precisely, we establish the global existence
of weak solutions for Navier-Stokes-BGK system consisting of the BGK model of
Boltzmann equation and incompressible Navier-Stokes equations coupled through a
drag forcing term. This is achieved by combining weak compactness of the
particle interaction operator based on Dunford-Pettis theorem, strong
compactness of macroscopic fields of the kinetic part relied on velocity
averaging lemma and a high order moment estimate, and strong compactness of the
fluid part by Aubin-Lions lemma
Large friction limit of pressureless Euler equations with nonlocal forces
We rigorously show a large friction limit of hydrodynamic models with
alignment, attractive, and repulsive effects. More precisely, we consider
pressureless Euler equations with nonlocal forces and provide a quantitative
estimate of large friction limit to a continuity equation with nonlocal
velocity fields, which is often called an aggregation equation. Our main
strategy relies on the relative entropy argument combined with the estimate of
-Wasserstein distance between densities
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