Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds

• Published in: Archiv der Mathematik 2024, Vol.122 (1), p.83-93
• Main Authors: Nguyen, Thieu Huy, Vu, Thi Ngoc Ha
• Format: Article
• Language: English
• Subjects: Curvature; Fields (mathematics); Fluid flow; Manifolds; Mathematical analysis; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Tensors
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-023-01923-5

Consider a noncompact Einstein manifold ( M ,  g ) with negative Ricci curvature tensor ( Ric ij = r g ij for a curvature constant r < 0 ). Denoting by Γ ( T M ) the set of all vector fields on M , we study the Navier–Stokes equations ∂ t u + ∇ u u + grad π = div ( ∇ u + ∇ u t ) ♯ , div u = 0 , u | t = 0 = u 0 ∈ Γ ( T M ) , div u 0 = 0 , for the vector field u ∈ Γ ( T M ) . Given any initial datum u 0 ∈ Γ ( T M ) , we prove that if the curvature constant r is large enough, then the Navier–Stokes equations on the Einstein manifold ( M ,  g ) always have a unique solution u ( · , t ) ∈ Γ ( T M ) which is defined for all t ≥ 0 with u ( · , 0 ) = u 0 . We also prove the exponential decay of solutions under appropriate conditions.