Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions

1993/01/01 by Geneviève Raugel, George R. Sell · 3 citations
Mathematics · Engineering · Computer Science · #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.1090/s0894-0347-1993-1179539-4

openalex publication_date 1993/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the Navier-Stokes equations (NS) on a thin <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript epsilon Baseline equals upper Q 2 times left-parenthesis 0 comma epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi> ε </mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo> × </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi> ε </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Ω _ε = Q₂ × (0,ε )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Q₂</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable bounded domain in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb R²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi> ε </mml:mi> <mml:annotation encoding="application/x-tex">ε</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a small, positive, real parameter. We consider these equations with various homogeneous boundary conditions, especially spatially periodic boundary conditions. We show that there are <italic>large</italic> sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R left-parenthesis epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ε </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal R(ε )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis normal upper Omega Subscript epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi> ε </mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H¹(Ω _ε )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S left-parenthesis epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ε </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal S(ε )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 1 comma normal infinity Baseline left-parenthesis left-parenthesis 0 comma normal infinity right-parenthesis comma upper L squared left-parenthesis normal upper Omega Subscript epsilon Baseline right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mm

Citations

• Nonautonomous differential equations and topological dynamics. I. The basic theory
• Sur le mouvement d'un liquide visqueux emplissant l'espace
• Nonautonomous differential equations and topological dynamics. II. Limiting equations

Cited by

• Navier-Stokes equations in thin 3D domains with Navier boundary conditions
• Navier–Stokes Equations in a Curved Thin Domain, Part II: Global Existence of a Strong Solution
• George Roger Sell [wikipedia]