Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping

We study the uniqueness, the continuity 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 , and the large time decay for the Leray solutions of the 3D incompressible Navier-Stokes equations with the nonlinear exponential damping term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>b</m:mi> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn mathvariant="bold">2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> </m:math> a({e}^{b| u{| }^{{2}}}-1)u , ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> a,b> 0 ).