scholarly journals Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhendong Fang ◽  
Hao Wang
Keyword(s):  
Initial Data ◽  
Knudsen Number ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Poisson System ◽  
Small Initial Data ◽  
Uniform Estimates ◽  
Poisson Boltzmann ◽  
Two Fluid

<p style='text-indent:20px;'>In this paper, we obtain the uniform estimates with respect to the Knudsen number <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> for the fluctuations <inline-formula><tex-math id="M2">\begin{document}$ g^{\pm}_{\varepsilon} $\end{document}</tex-math></inline-formula> to the two-species Vlasov-Poisson-Boltzmann (in briefly, VPB) system. Then, we prove the existence of the global-in-time classical solutions for two-species VPB with all <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon \in (0,1] $\end{document}</tex-math></inline-formula> on the torus under small initial data and rigorously derive the convergence to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system as <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> go to 0.</p>

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

scholarly journals Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

scholarly journals L1STABILITY FOR THE VLASOV–POISSON–BOLTZMANN SYSTEM AROUND VACUUM

2006 ◽  
Vol 16 (09) ◽  
pp. 1505-1526 ◽  
Author(s):  
RENJUN DUAN ◽  
MEI ZHANG ◽  
CHANGJIANG ZHU
Keyword(s):  
Existence Theory ◽  
Classical Solutions ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Small Initial Data ◽  
Dimensional Phase Space ◽  
Nonlinear Functionals ◽  
Poisson Boltzmann ◽  
The Stability ◽  
Hard Potentials

Based on the global existence theory of the Vlasov–Poisson–Boltzmann system around vacuum in the N-dimensional phase space, in this paper, we prove the uniform L1stability of classical solutions for small initial data when N ≥ 4. In particular, we show that the stability can be established directly for the soft potentials, while for the hard potentials and hard sphere model it is obtained through the construction of some nonlinear functionals. These functionals thus generalize those constructed by Ha for the case without force to capture the effect of the force term on the time evolution of solutions. In addition, the local-in-time L1stability is also obtained for the case of N = 3.

scholarly journals Fujita–Kato solution for compressible Navier–Stokes equations with axisymmetric initial data and zero Mach number limit

2019 ◽  
Vol 22 (05) ◽  
pp. 1950041
Author(s):  
Boris Haspot
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Dimension Formula ◽  
Global Strong Solution ◽  
Rotational Part ◽  
Zero Mach Number Limit

In this paper, we investigate the question of the existence of global strong solution for the compressible Navier–Stokes equations for small initial data such that the rotational part of the velocity [Formula: see text] belongs to [Formula: see text] (in dimension [Formula: see text]). We show then an equivalent of the so-called Fujita–Kato theorem to the case of the compressible Navier–Stokes equations when we consider axisymmetric initial data. The main difficulty is linked to the fact that in this case the velocity is not Lipschitz, as a consequence we have to study carefully the coupling between the rotational and irrotational part of the velocity. In a second part, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero.

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