scholarly journals On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

2021 ◽  
Author(s):  
Shi Jin ◽  
Esther Daus ◽  
Liu Liu
Keyword(s):  
Boltzmann Equation ◽  
Galerkin Approximation ◽  
Single Species ◽  
Collision Kernel ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time ◽  
Stochastic Galerkin ◽  
Analytical Tools ◽  
Gap Estimate

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, Arch. Ration. Mech. Anal., 3, 1367–1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, Kinet. Relat. Models, 12, 909–922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

scholarly journals Well-posedness and long-time behavior for the modified phase-field crystal equation

2014 ◽  
Vol 24 (14) ◽  
pp. 2743-2783 ◽  
Author(s):  
Maurizio Grasselli ◽  
Hao Wu
Keyword(s):  
Phase Field ◽  
Time Derivative ◽  
Exponential Attractor ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Global Existence And Uniqueness ◽  
And Diffusion ◽  
Phase Field Crystal

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K. R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the global existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the long-time behavior of the corresponding dynamical system. In particular, we establish the existence of the global attractor as well as an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space converges to a single equilibrium. This is done by means of a suitable version of the Łojasiewicz–Simon inequality. An estimate on the convergence rate is also given.

scholarly journals Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

2014 ◽  
Vol 12 ◽  
pp. 10-20
Author(s):  
Jiang Bo Zhou ◽  
Jun De Chen ◽  
Wen Bing Zhang
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Holm Equation ◽  
Special Cases ◽  
Long Time

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

scholarly journals Well-posedness and long-time behavior for a class of doubly nonlinear equations

2007 ◽  
Vol 18 (1) ◽  
pp. 15-38 ◽  
Author(s):  
Giulio Schimperna ◽  
◽  
Antonio Segatti ◽  
Ulisse Stefanelli ◽  
◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Doubly Nonlinear

scholarly journals An approach without using Hardy inequality for the linear heat equation with singular potential

2015 ◽  
Vol 17 (05) ◽  
pp. 1550041 ◽  
Author(s):  
Lucas C. F. Ferreira ◽  
Cláudia Aline A. S. Mesquita
Keyword(s):  
Heat Equation ◽  
Hardy Inequality ◽  
Threshold Value ◽  
Energy Estimates ◽  
Well Posedness ◽  
Time Behavior ◽  
Linear Heat Equation ◽  
Linear Heat ◽  
Long Time ◽  
The Fourier Transform

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation ut - Δu - V(x)u = 0 in ℝn with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely, PMk-spaces. For arbitrary data u0 ∈ PMk, the approach allows to compute an explicit smallness condition on V for global existence in the case of V with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the case of the monopolar potential [Formula: see text] with [Formula: see text]. This threshold value is the same one obtained for the global well-posedness of L2-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between L2 and PMk, our results indicate that λ* is intrinsic of the PDE and independent of a particular approach. We also analyze the long-time behavior of solutions and show there are infinitely many possible asymptotics characterized by the cells of a disjoint partition of the initial data class PMk.

ENERGY CONSERVING DISCRETE BOLTZMANN EQUATION FOR NONIDEAL SYSTEMS

1999 ◽  
Vol 10 (07) ◽  
pp. 1367-1382 ◽  
Author(s):  
NICOS S. MARTYS
Keyword(s):  
Boltzmann Equation ◽  
Autocorrelation Function ◽  
Collision Operator ◽  
Lattice Boltzmann Model ◽  
Nonlocal Interaction ◽  
Time Behavior ◽  
Moment Equations ◽  
Long Time ◽  
Energy Conserving ◽  
Discrete Boltzmann Equation

The BBGKY formalism is utilized to obtain a set of moment equations to be satisfied by the collision operator in an energy conserving discrete Boltzmann equation for the case of a nonlocal interaction potential. A modified BGK form of the collision operator consistent with these moment equations is described. In the regime of isothermal flows, a previous proposed nonideal gas model is recovered. Other approaches to constructing the collision operator are discussed. Numerical implementation of the modified BGK form, using a thermal lattice Boltzmann model, is illustrated as an example. The time dependence of the density autocorrelation function was studied for this model and found, at early times, to be strongly affected by the constraint of total energy conservation. The long time behavior of the density autocorrelation function was consistent with the theory of hydrodynamic fluctuations.

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