A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

By Raphaël Danchin, Piotr Bogusław Mucha. 2011

217 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77dfffbac59ee01ed5e5546eda7fa2526e67760c

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Citations in the corpus

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity (2012, methodology): “Very recently, Danchin and Mucha [13] filled the gap for the uniqueness result in [1] with p ∈ (d, 2d) through Langrage approach, and Abidi, Gui and Zhang relaxed the smalness condition for a0 in [4, 5].”
Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system (2012, background): “[12] R. Danchin, P.B. Mucha, A Lagrangian approach for the incompressible Navier–Stokes equations with variable density, Comm. Pure Appl.”
On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium (2015, background): “1 in [14], where the authors proved (2.”
Global solutions to 2-d inhomogeneous navier-stokes system with general velocity (2012, methodology): “techniques in [10] to prove the global existence of the solution to (1.”
On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces (2014, background): “ρ,u,K) to (1.8) with ρ∈ C([0,T];B˙ n p p,1) and (u,K) ∈ Ep(T). Then one may prove that, under Condition (3.15), the “ﬂow” X(t,) of (˙ u,K) deﬁned by (1.10) is a C1-diﬀeomorphism over Rn (see [10] and [11]), and satisﬁes DX− Id ∈ WELL-POSEDNESS OF FULL COMPRESSIBLE NAVIER-STOKES SYSTEM 21 C([0,T];B˙ ”
Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density (2013, methodology): “We emphasize that the Lagrangian idea introduced in [9, 10] will also be essential for the proof of the uniqueness result here.”