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

Abstract This work focuses on the study of non-autonomous incompressible third-grade fluid equations in two- and three-dimensional open and connected domains $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> , 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 $$\Psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> . Second, on bounded domains, we prove the existence of a unique global pullback attractor (GPA) for the underlying system using the compact Sobolev embedding $${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>↪</mml:mo> <mml:mo>↪</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Lastly, we demonstrate the existence of a unique GPA by considering the underlying system on unbounded Poincaré domains. Since the compact Sobolev embedding $${\mathbb {H}}^1(\Omega )\hookrightarrow \hookrightarrow {\mathbb {L}}^2(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>↪</mml:mo> <mml:mo>↪</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\Psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> . We notice that the pullback asymptotic compactness of $$\Psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> 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.