Global well-posedness to the subcritical Oldroyd-B type models in 2D

Abstract

We prove the global well-posedness to the 2D Oldroyd-B type models with $\nu {\Lambda }^{2\alpha }u$ and $\eta {\Lambda }^{2\beta }\tau$ satisfying $(i)\alpha >1,\eta =0$ or $(ii)\alpha =1,\beta >0$. By establishing the gradient estimate of $u$, $\tau$ and $L^{\infty }$ bound of $\mathrm{curlu}+{\mathrm{\Lambda }}^{-2}\mathrm{curldiv}\tau$, Elgidi-Rousset (Commun. Pure Appl. Math. online, 2015) obtained the global well-posedness for the case $\nu =0$, $\beta =1$. However, for the cases $(i)$ and $(ii)$, it is difficult to improve the regularity of $u$ and $\tau$ directly, especially when $\alpha \to 1^{+}$ in case $(i)$ and $\beta \to 0^{+}$ in case $(ii)$. To overcome this difficulty, we exploit a new structure of the equations coming from the dissipation and coupled term. Then we prove the global well-posedness to these cases by energy method which brings us closer to the more interesting case $\alpha =1$, $\eta =0$.

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References in corpus (2)

• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• The 2D incompressible Boussinesq equations with general critical dissipation