A Low Mach Number Limit of a Dispersive Navier–Stokes System

2012 ◽  
Vol 44 (3) ◽  
pp. 1760-1807 ◽  
Author(s):  
C. David Levermore ◽  
Weiran Sun ◽  
Konstantina Trivisa
Keyword(s):  
Mach Number ◽  
Stokes System ◽  
Navier Stokes ◽  
Low Mach Number ◽  
Low Mach Number Limit

scholarly journals ON THE WELL-POSEDNESS OF THE FULL LOW MACH NUMBER LIMIT SYSTEM IN GENERAL CRITICAL BESOV SPACES

2012 ◽  
Vol 14 (03) ◽  
pp. 1250022 ◽  
Author(s):  
RAPHAËL DANCHIN ◽  
XIAN LIAO
Keyword(s):  
Mach Number ◽  
Besov Spaces ◽  
Stokes System ◽  
Initial Temperature ◽  
Navier Stokes ◽  
Well Posedness ◽  
Low Mach Number ◽  
Limit System ◽  
Low Mach Number Limit ◽  
Local Existence And Uniqueness

This work is devoted to the well-posedness issue for the low Mach number limit system obtained from the full compressible Navier–Stokes system, in the whole space ℝd with d ≥ 2. In the case where the initial temperature (or density) is close to a positive constant, we establish the local existence and uniqueness of a solution in critical homogeneous Besov spaces of type [Formula: see text]. If, in addition, the initial velocity is small then we show that the solution exists for all positive time. In the fully nonhomogeneous case, we establish the local well-posedness in nonhomogeneous Besov spaces [Formula: see text] (still with critical regularity) for arbitrarily large data with positive initial temperature. Our analysis strongly relies on the use of a modified divergence-free velocity which allows to reduce the system to a nonlinear coupling between a parabolic equation and some evolutionary Stokes system. As in the recent work by Abidi and Paicu [Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57(3) (2007) 883–917]. Concerning the density-dependent incompressible Navier–Stokes equations, the Lebesgue exponents of the Besov spaces for the temperature and the (modified) velocity, need not be the same. This enables us to consider initial data in Besov spaces with a negative index of regularity.

scholarly journals On the Low Mach Number Limit for Quantum Navier--Stokes Equations

2020 ◽  
Vol 52 (6) ◽  
pp. 6105-6139
Author(s):  
Paolo Antonelli ◽  
Lars Eric Hientzsch ◽  
Pierangelo Marcati
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Low Mach Number Limit

Singular limit of a dispersive Navier–Stokes system with an entropy structure

2014 ◽  
Vol 13 (01) ◽  
pp. 77-99
Author(s):  
C. David Levermore ◽  
Weiran Sun ◽  
Konstantina Trivisa
Keyword(s):  
Stokes System ◽  
Singular Limit ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Third Order ◽  
Fluid System ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Whole Space ◽  
Entropy Structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

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