On the Asymptotic Behavior in Time of the Kinetic Energy in a Rigid Body–Liquid Problem

Sufficient conditions on the initial data are given for the decay in time of the kinetic energy, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , of solutions to the system of equations describing the motion of a rigid body in a Navier–Stokes liquid. More precisely, under the assumption that the initial data are “small” in an appropriate norm, it is shown that if, in addition, the initial velocity field of the liquid, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v 0"> <mml:semantics> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">v_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q element-of left-parenthesis 1 comma 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q\in (1,2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a specific order of decay. The order remains, however, unspecified if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v 0 element-of upper L squared"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">v_0\in L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .