<p>This work focuses on the study of non-autonomous incompressible third-grade fluid equations in two- and three-dimensional open and connected domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, which may be either bounded or unbounded. Our main goals are to investigate the well-posedness of the equations and to analyze the asymptotic behavior of their solutions. First, we establish the existence of at least one weak solution for the system with the Dirichlet boundary condition. Next, we show that every weak solution satisfies the energy equality, ensuring continuity over time. This, in turn, helps to establish the uniqueness of weak solutions and guarantees that the underlying system can be associated with a continuous cocycle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>. Second, on bounded domains, we prove the existence of a unique global pullback attractor (GPA) for the underlying system using the compact Sobolev embedding <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">↪</mo> <mo stretchy="false">↪</mo> <msup> <mrow> <mi mathvariant="double-struck">L</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Lastly, we demonstrate the existence of a unique GPA by considering the underlying system on unbounded Poincaré domains. Since the compact Sobolev embedding <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">↪</mo> <mo stretchy="false">↪</mo> <msup> <mrow> <mi mathvariant="double-struck">L</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is no longer available in this setting, we employ the uniform-tail estimates approach to obtain the existence of a unique GPA on unbounded domains. This allows us to show the pullback asymptotic compactness of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>. We notice that the pullback asymptotic compactness of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation> in unbounded domains cannot be obtained using the energy equality method due to the presence of strong nonlinear terms in the equations, which substantially complicates the analysis, making the study of unbounded domains more challenging. The main novelty of this work, compared to the existing literature, is that our results yield solutions that satisfy energy equality and are continuous in time. Moreover, to the best of the authors’ knowledge, this is the first work that considers 2D as well as 3D non-autonomous incompressible third-grade fluids and establishes the existence of a unique GPA in bounded as well as unbounded domains.</p>

Well-posedness and asymptotic analysis of a class of 2D and 3D third-grade fluids in bounded and unbounded domains

期刊论文
出版模式：

开放获取
发表日期：

2025年6月27日

查看全文

Kush Kinra,
Fernanda Cipriano

摘要

This work focuses on the study of non-autonomous incompressible third-grade fluid equations in two- and three-dimensional open and connected domains $\Omega $ $Ω$ , which may be either bounded or unbounded. Our main goals are to investigate the well-posedness of the equations and to analyze the asymptotic behavior of their solutions. First, we establish the existence of at least one weak solution for the system with the Dirichlet boundary condition. Next, we show that every weak solution satisfies the energy equality, ensuring continuity over time. This, in turn, helps to establish the uniqueness of weak solutions and guarantees that the underlying system can be associated with a continuous cocycle $\Psi $ $Ψ$ . Second, on bounded domains, we prove the existence of a unique global pullback attractor (GPA) for the underlying system using the compact Sobolev embedding ${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$ $H^{1} (Ω) ↪ ↪ L^{2} (Ω)$ . Lastly, we demonstrate the existence of a unique GPA by considering the underlying system on unbounded Poincaré domains. Since the compact Sobolev embedding ${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$ $H^{1} (Ω) ↪ ↪ L^{2} (Ω)$ is no longer available in this setting, we employ the uniform-tail estimates approach to obtain the existence of a unique GPA on unbounded domains. This allows us to show the pullback asymptotic compactness of $\Psi $ $Ψ$ . We notice that the pullback asymptotic compactness of $\Psi $ $Ψ$ in unbounded domains cannot be obtained using the energy equality method due to the presence of strong nonlinear terms in the equations, which substantially complicates the analysis, making the study of unbounded domains more challenging. The main novelty of this work, compared to the existing literature, is that our results yield solutions that satisfy energy equality and are continuous in time. Moreover, to the best of the authors’ knowledge, this is the first work that considers 2D as well as 3D non-autonomous incompressible third-grade fluids and establishes the existence of a unique GPA in bounded as well as unbounded domains.