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A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density
Author(s) -
Danchin Raphaël,
Mucha Piotr Bogusław
Publication year - 2012
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21409
Subject(s) - mathematics , piecewise , uniqueness , mathematical analysis , constant (computer programming) , jump , compressibility , lagrangian and eulerian specification of the flow field , lagrangian , eulerian path , physics , quantum mechanics , computer science , thermodynamics , programming language
We investigate the Cauchy problem for the inhomogeneous Navier‐Stokes equations in the whole n ‐dimensional space. Under some smallness assumption on the data, we show the existence of global‐in‐time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p‐1}_{p,1}({\Bbb R}^n)$ . In particular, piecewise‐constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.