Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier‐Stokes equations supplemented with H1 initial velocity and only bounded nonnegative density. In contrast to all the previous works on those topics, we do not require regularity or a positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions. Those solutions are global in the two‐dimensional case for general data, and in the three‐dimensional case if the velocity satisfies a suitable scaling‐invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in his 1996 book Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models, concerning the evolution of a drop of incompressible viscous fluid in the vacuum. © 2018 Wiley Periodicals, Inc.

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“…The main issue is to establish that the smoothness of D 0 is preserved through the time evolution (see [10,13,21,22]).…”

Section: Introductionmentioning

confidence: 99%

The Incompressible Navier‐Stokes Equations in Vacuum

Danchin¹,

Mucha

2018

Comm Pure Appl Math

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“…When one assumes that the viscous coefficient is a positive constant, Liao and the second author [20,21] solved the case when the system (1.1) is supplemented with the initial density, ρ 0 (x) = η 1 1 Ω 0 + η 2 1 Ω c 0 , for some pair of positive constants (η 1 , η 2 ), and for any bounded, simply connected domain Ω 0 with W k+2,p (R 2 ) (p ∈]2, 4[) boundary regularity. Danchin and Zhang [12] and Gancedo and Garcia-Juarez [15] proved the propagation of C k+γ regularity of the interface for k = 1 or k = 2. Lately Danchin and Mucha [11] proved the propagation of C 1+γ regularity of density patch which allows vacuum.…”

Section: Introductionmentioning

confidence: 99%

Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

Paicu

Zhang

2019

Commun. Math. Phys.

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In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the L 1 in time Lipschitz estimate of the velocity field can not be obtained by energy method (see [11,20,21] for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid ([6, 7]), namely, striated regularity can help to get the L ∞ boundedness of the double Riesz transform, we derive the a priori L 1 in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of H 3 regularity of the interface between fluids with different densities.

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“…Acknowledgment. After this paper was finished, we found that there are several papers on the propagation of C kC regularity of the density patches for viscous inhomogeneous incompressible fluid flow (see [16,18]).…”

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confidence: 99%

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

Liao

Zhang

2018

Comm Pure Appl Math

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This paper presents some progress toward an open question proposed by P.‐L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global‐in‐time well‐posedness of the two‐dimensional inhomogeneous incompressible Navier‐Stokes system with initial density ρ0=η1boldnormal1Ω 0+η2boldnormal1Ω0c. Here (η1,η2) is any pair of positive constants and Ω0 is a bounded, simply connected W3,p(ℝ2) domain. We then prove that for any positive time t, the density ρ(t)=η1boldnormal1Ω(t)+η2boldnormal1Ω(t)c, with the domain Ω(t) preserving the W3,p‐boundary regularity. © 2018 Wiley Periodicals, Inc.

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Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

Cited by 25 publications

References 44 publications

The Incompressible Navier‐Stokes Equations in Vacuum

The Incompressible Navier‐Stokes Equations in Vacuum

Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

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