Incompressible Flows with Piecewise Constant Density

Danchin, Raphaël; Mucha, Piotr B.

doi:10.1007/s00205-012-0586-4

Cited by 84 publications

(110 citation statements)

References 25 publications

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“…Furthermore, with the initial density fluctuation being sufficiently small, for any initial velocity v 0 2 B 1 4;2 .R 2 / \ L 2 .R 2 / or v 0 2 B 2 2=q q;p .R d / with small size for 1 < p < 1, d < q < 1, and 2 2 p ¤ 1 q ; they also proved the global well-posedness of (1.1). M. Paicu, Z. Zhang, and the second author [27] improved the well-posedness results in [14] with less regularity assumptions on the initial velocity.…”

Section: Introductionmentioning

confidence: 99%

“…In the general case where 0 2 L 1 .R d / with a positive lower bound and v 0 2 H 2 .R d /; R. Danchin and P.-B. Mucha [14] proved that the system (1.1) has a unique local-in-time solution. Furthermore, with the initial density fluctuation being sufficiently small, for any initial velocity v 0 2 B 1 4;2 .R 2 / \ L 2 .R 2 / or v 0 2 B 2 2=q q;p .R d / with small size for 1 < p < 1, d < q < 1, and 2 2 p ¤ 1 q ; they also proved the global well-posedness of (1.1).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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“…Here one can mention the critical regularity approach of [4,5] where density has to be continuous, bounded, and bounded away from 0, and relatively new works like [6,8] (further improved in [3,9,15,29]), relying on the use of Lagrangian coordinates, and where the density need not be continuous. Recently, in connection with Lions' question, much attention has been brought to the case where the initial density is given by…”

Section: Introductionmentioning

confidence: 99%

The Incompressible Navier‐Stokes Equations in Vacuum

Danchin¹,

Mucha

2018

Comm Pure Appl Math

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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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“…This result improves the former interesting well-posedness theorem of Danchin and Mucha in [14] by removing the smallness assumption on the fluctuation to the initial density and also with much less regularity for the initial velocity.…”

Section: Introductionmentioning

confidence: 49%

On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient

Abidi

Zhang

2015

Journal of Differential Equations

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Given solenoidal vector u 0 ∈Ḣ −2δ ∩ H 1 (R 2 ), ρ 0 − 1 ∈ L 2 (R 2 ), and ρ 0 ∈ L ∞ ∩Ẇ 1,r (R 2 ) with a positive lower bound for δ ∈ (0, 1 2 ) and 2 < r < 2 1−2δ , we prove that 2-D incompressible inhomogeneous Navier-Stokes system (1.1) has a unique global solution provided that the viscous coefficient μ(ρ 0 ) is close enough to 1 in the L ∞ norm compared to the size of δ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for 1 < p < 4, if (ρ 0 − 1, u 0 ) belongs to the critical Besov spaces Ḃ 2 p p,1 (R 2 ) × Ḃ −1+ 2 p p,1 ∩ L 2 (R 2 ) and the Ḃ 2 p p,1 (R 2 ) norm of ρ 0 − 1 is sufficiently small compared to the exponential of u 0 2 L 2 + u 0 Ḃ −1+ 2 p p,1 , we prove the global well-posedness of (1.1) in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of (1.1) under the assumption that the L ∞ norm of ρ 0 − 1 is sufficiently small.

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Incompressible Flows with Piecewise Constant Density

Cited by 84 publications

References 25 publications

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

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

The Incompressible Navier‐Stokes Equations in Vacuum

On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient

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