A class of global large solutions to the compressible Navier–Stokes–Korteweg system in critical Besov spaces

Zhang, Shunhang

doi:10.1007/s00028-020-00565-2

Cited by 9 publications

(3 citation statements)

References 23 publications

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“…Following the assumptions on the viscosity coefficients of Haspot, 18 Zhai and Li 35 recently proved the global solutions to system () without smallness condition imposed on the vertical component of the incompressible part of the velocity by using the weighted Chemin‐Lerner‐norm technique. Zhang 36 constructed a class of global large solutions to the compressible NSK system with constant viscosity coefficients in critical Besov spaces. Recently, Yu et al 37 constructed global smooth solutions to system () with a class of special initial data, where the initial velocity in

L^{\infty} false(ℝ^{3} false)

can be arbitrarily large while the initial data ρ 0 are small in

H^{3} false(ℝ^{3} false)

.…”

Section: Introductionmentioning

confidence: 99%

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

Yang

2022

Math Methods in App Sciences

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L^{\infty} false(ℝ^{3} false)

can be arbitrarily large while the initial data ρ 0 are small in

H^{3} false(ℝ^{3} false)

.…”

Section: Introductionmentioning

confidence: 99%

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

Yang

2022

Math Methods in App Sciences

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“…In the present paper, the ρ * has been set to 1 without loss of generality. Furthermore, as recent contributions, we refer to the following papers: Zhang [39]; Zhai and Li [38]; Kawashima, Shibata, and Xu [15]; Charve, Danchin, and Xu [4].…”

Section: Introductionmentioning

confidence: 99%

Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

Kobayashi

Murata

Saito

2021

J. Math. Fluid Mech.

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In this paper, we consider a resolvent problem arising in the study of a Korteweg-type model of isothermal compressible viscous fluids derived by Dunn and Serrin (1985), and we prove resolvent estimates in bounded and exterior domains. The Korteweg-type model is employed to describe fluid capillarity effects or a liquid-vapor two-phase flow with phase transition as diffuse interface model. Our resolvent problem is obtained from a linearized system of the Korteweg-type model around the steady state (ρ, u) = (1, 0), where ρ is the fluid density and u is the fluid velocity. For bounded domains, we treat variable coefficients and prove that the resolvent set contains C + = {z ∈ C : z ≥ 0} under the condition that the pressure function P (ρ) satisfies not only P (1) ≥ 0 but also P (1) < 0, where P (1) = (dP/dρ)(1). We follow the idea of Kotschote (2014) to handle P (1) < 0. For exterior domains, we treat constant coefficients and prove that the resolvent set contains {z ∈ C + : |z| ≥ δ} for any δ > 0 when P (1) ≥ 0. Furthermore, we introduce a global solvability result for the nonlinear problem in a bounded domain as an application of the resolvent estimate.

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“…In [49], Watanabe proved the existence of a unique global strong solutions to the compressible Navier-Stokes-Korteweg equations in the L p -in-time and L q -inspace framework. Readers interested by Korteweg type systems are referred to the following articles [1,2,3,4,5,6,8,9,21,29,36,47,42,53] and references cited therein.…”

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confidence: 99%

Global large solutions and optimal time-decay estimates to the Korteweg system

Zhai¹,

Li²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

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A class of global large solutions to the compressible Navier–Stokes–Korteweg system in critical Besov spaces

Cited by 9 publications

References 23 publications

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

Global smooth solutions of 3‐D Navier‐Stokes‐Korteweg equations with large initial data

Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

Global large solutions and optimal time-decay estimates to the Korteweg system

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