The Optimal Time Decay Rates and Vanishing Limit for Three-Dimensional Incompressible Phan-Thien–Tanner and Oldroyd-B Systems in the Critical Besov Spaces

In this article, we study the incompressible Phan-Thien–Tanner (PTT) and Oldroyd-B systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . The interesting features of the Cauchy problem studied in this work are the well-posedness and large-time behavior of global solutions for the PTT and Oldroyd-B systems with or without a damping mechanism, and the relationship between these two systems in the critical Besov spaces. First, we study the well-posedness of global solutions with data having critical regularity in the case of small initial data by proving uniform estimates with respect to the parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>a</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>b</mml:mi> </mml:math> . Second, we prove the optimal time decay rates of global solutions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> by exploiting harmonic analysis tools, such as non-standard <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> product estimates, various Sobolev embeddings, and interpolation inequalities. We only assume that the negative Besov norm at low frequencies of the initial data is bounded and remove the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:math> ) condition, which has been required in previous works on this problem. Finally, we investigate the convergence of global solutions to the PTT system when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>b</mml:mi> </mml:math> tends to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>0</mml:mn> </mml:math> . Moreover, the specific rate of convergence is obtained in some sense.