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<p>We consider the Cauchy problem (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d, d=2,3\)</EquationSource> </InlineEquation>) and the initial periodic-boundary values problem (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {T}^d, d=2,3\)</EquationSource> </InlineEquation>) associated to the compressible Oldroyd-B model which is first derived by Barrett, Lu and Süli [Commun. Math. Sci., 15 (2017)] through micro–macro-analysis of the compressible Navier-Stokes-Fokker-Planck system. The lack of stress diffusion causes the problems hard to study. Exploiting tools from harmonic analysis, particularly the Littlewood-Paley theory, we establish the global well-posedness and time-decay rates for solutions of the Cauchy problem with small initial data in Besov spaces with critical regularity. By deeply exploring the structure of the perturbation system, we also obtain the global well-posedness and exponential decay rates for solutions of the initial periodic-boundary value problem with small initial data in the Sobolev spaces. We improve the recent results by Lu and Pokorný [Anal. Theory Appl., 36 (2020)], Wang and Wen [Math. Models Methods Appl. Sci., 30 (2020)], and Liu, Lu and Wen [SIAM J. Math. Anal., 53 (2021)].</p>

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Global well-posedness and large time behavior of solutions to the compressible Oldroyd-B model without stress diffusion

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Yajuan Zhao,
  • Yongsheng Li,
  • Tao Liang,
  • Xiaoping Zhai

摘要

We consider the Cauchy problem ( \(\mathbb {R}^d, d=2,3\) ) and the initial periodic-boundary values problem ( \(\mathbb {T}^d, d=2,3\) ) associated to the compressible Oldroyd-B model which is first derived by Barrett, Lu and Süli [Commun. Math. Sci., 15 (2017)] through micro–macro-analysis of the compressible Navier-Stokes-Fokker-Planck system. The lack of stress diffusion causes the problems hard to study. Exploiting tools from harmonic analysis, particularly the Littlewood-Paley theory, we establish the global well-posedness and time-decay rates for solutions of the Cauchy problem with small initial data in Besov spaces with critical regularity. By deeply exploring the structure of the perturbation system, we also obtain the global well-posedness and exponential decay rates for solutions of the initial periodic-boundary value problem with small initial data in the Sobolev spaces. We improve the recent results by Lu and Pokorný [Anal. Theory Appl., 36 (2020)], Wang and Wen [Math. Models Methods Appl. Sci., 30 (2020)], and Liu, Lu and Wen [SIAM J. Math. Anal., 53 (2021)].