On the global regularity of the 2D Oldroyd-B-type model
The author studies the classical Oldroyd-B model with diffusive stress, introduced in [\textit{J. G. Oldroyd}, Proc. R. Soc. Lond., Ser. A 245, 278--297 (1958; Zbl 0080.38805)]. The Zygmund operators $(-\Delta)^{\alpha},(-\Delta)^{\gamma} $, defined via the Fourier transform, appear in the equations for velocity and for the non-Newtonian part of the stress tensor. The global regularity of the 2D Oldroyd-B model for suitably small initial $L^2$-energy norm is proven in the paper for two cases: (a) $\alpha + \beta = 2$ with $0 < \beta < 2$; (b) the borderline case $\alpha = 0$, $\beta = 2$. Some technical methods given in [\textit{P. Constantin} and \textit{M. Kliegl}, Arch. Ration. Mech. Anal. 206, No. 3, 725--740 (2012; Zbl 1366.76006)] and [\textit{T. Kato} and \textit{G. Ponce}, Commun. Pure Appl. Math. 41, No. 7, 891--907 (1988; Zbl 0671.35066)] are adapted and used to obtain these results. \par The main tools are the Young and Gronwall inequalities, the logarithmic-type Sobolev inequality, the log-Gronwall inequality, the frequency operator and the Plancherel theorem. The Bernstein-type inequalities for fractional derivatives are also used, and a brief description of the basic functional spaces used in this paper -- the Besov spaces -- is given in the appendix. The important restriction $0 \leq \beta \leq 1$ implies an estimate of the vorticity in terms of initial data, which allow to obtain the estimate of $\nabla u$ with respect to initial data.
scientific article; zbMATH DE number 7041960
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