<p>We investigate the role of the four viscosity parameters in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, in the case of a spin viscosity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and we construct the asymptotic profiles of the solution as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t\rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, and not on the other parameters <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> (vortex-viscosity), <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> (spin viscosity) and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times.</p>

On the Role of the Viscosity Parameters in the Large Time Asymptotics of Two Dimensional Micropolar Flows

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2026年5月21日

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L. Brandolese,
A. V. Busuioc,
D. Iftimie,
C. F. Perusato

摘要

We investigate the role of the four viscosity parameters in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in $L^2$ $L^{2}$ , in the case of a spin viscosity $\gamma \ge 0$ $γ \geq 0$ , and we construct the asymptotic profiles of the solution as $t\rightarrow +\infty $ $t \to + \infty$ . We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity $\mu $ $μ$ , and not on the other parameters $\chi $ $χ$ (vortex-viscosity), $\gamma $ $γ$ (spin viscosity) and $\kappa $ $κ$ (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times.