Optimal large time behavior of the 3D rate type viscoelastic fluids

Abstract We investigate optimal decay estimates of solutions to the 3D Cauchy problem of the rate type viscoelastic fluids. The main novelty of this article involves three aspects: first, we show that the second-order and third-order spatial derivative of the global solution converges to zero at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}^{2} -rate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>7</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:math> {\left(1+t)}^{-\tfrac{7}{4}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>9</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:math> {\left(1+t)}^{-\tfrac{9}{4}} , respectively, which improve previous related decay results. Second, we prove that the decay rate of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m th order spatial derivative (where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo form="prefix">,</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> m\in {[}0,3] ) of the extra stress tensor of the field in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}^{2} is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>5</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:mstyle> <m:mo>−</m:mo> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:math> {\left(1+t)}^{-\tfrac{5}{4}-\tfrac{m}{2}} , which is faster than that of the velocity. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates, which coincide with the upper rates. Therefore, our results are precisely optimal in this sense.