Global existence of classical solutions for the 3D generalized compressible Oldroyd‐B model

Zhu, Yi

doi:10.1002/mma.6393

Cited by 4 publications

(8 citation statements)

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“…It seems impossible to get the dissipative estimate of ∇ 4 u in the non-viscous case. To handle the issue, inspired by [52], we use the variation of the continuity equation divu = − ρt+βu•∇ρ r1+βρ and integration by parts to transfer the derivative to other term. Concerning the optimal time-decay estimates, the loss of dissipation of velocity due to the vanishing of viscosity is the main difficulty compared with the viscus case in [47].…”

Section: Resultsmentioning

confidence: 99%

“…Very recently, Zhou, Zhu and Zi ( [51]) obtained some time-decay estimates of strong solutions. Zhu [52] obtained the global well posedness of small classical solutions to a generalized inviscid compressible Oldroyd-B model in Sobolev space H s for s ≥ 5. In [3], Barrett, Lu and Süli not only showed the derivation of the compressible viscous Oldroyd-B model with stress diffusion (1.1) via a macroscopic closure of a micro-macro model, but also proved the existence of global-in-time finiteenergy weak solutions with arbitrarily large initial data in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

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The Cauchy problem for an inviscid Oldroyd-B model in $\mathbb{R}^3$

Liu,

Wang,

Wen

2021

Preprint

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In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with our previous work [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. In addition, it proves that the decay rates for the highest-order derivatives of the solutions are optimal. Our proof relies on Fourier theory and delicate energy method. This work can be viewed as an extension of [47].

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for an inviscid Oldroyd-B model in $\mathbb{R}^3$

Liu,

Wang,

Wen

2021

Preprint

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“…Very recently, Zhou, Zhu and Zi [51] obtained some time-decay estimates of strong solutions. Zhu [52] obtained the global well posedness of small classical solutions to a generalized inviscid compressible Oldroyd-B model in Sobolev space for . In [3], Barrett, Lu and Süli not only showed the derivation of the compressible viscous Oldroyd-B model with stress diffusion (1.1) via a macroscopic closure of a micro-macro model, but also proved the existence of global-in-time finite-energy weak solutions with arbitrarily large initial data in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for an inviscid Oldroyd-B model in three dimensions: global well posedness and optimal decay rates

Liu¹,

Wang²,

Wen³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method.

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“…When ν 1 = 0, (1) is also called the generalized compressible Oldroyd-B model. Zhu [32] investigated the three dimensional generalized compressible Oldroyd-B model and prove global existence of classical solutions.…”

mentioning

confidence: 99%

“…But global classical solutions to the problem (5), (6) in two space dimensions are still not obtained. In fact, global classical solution in two space dimensions may be established by using the similar method in [32].…”

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confidence: 99%

Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model

Liu

Wang

2022

EECT

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<p style='text-indent:20px;'>In this paper, we investigate the optimal time-decay rates of global classical solutions for the compressible Oldroyd-B model in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n(n = 2,3) $\end{document}</tex-math></inline-formula>. Global classical solutions in two space dimensions are still open. We first complete the proof of global classical solutions in two space dimensions. Based on global classical solutions and Fourier spectrum analysis, we obtain the optimal time-decay rates of global classical solutions in two and three space dimensions. More precisely, if the initial data belong to <inline-formula><tex-math id="M2">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>, the optimal time-decay rate of solutions and time-decay rates of <inline-formula><tex-math id="M3">\begin{document}$ l(l = 1,\cdot\cdot\cdot,m) $\end{document}</tex-math></inline-formula> order derivatives under additional assumptions are established.</p>

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Global existence of classical solutions for the 3D generalized compressible Oldroyd‐B model

Cited by 4 publications

References 42 publications

The Cauchy problem for an inviscid Oldroyd-B model in $\mathbb{R}^3$

The Cauchy problem for an inviscid Oldroyd-B model in $\mathbb{R}^3$

The Cauchy problem for an inviscid Oldroyd-B model in three dimensions: global well posedness and optimal decay rates

Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model

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