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

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...

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Published in:Archiv der Mathematik 2024, Vol.122 (1), p.83-93
Main Authors: Nguyen, Thieu Huy, Vu, Thi Ngoc Ha
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description 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.
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subjects Curvature
Fields (mathematics)
Fluid flow
Manifolds
Mathematical analysis
Mathematics
Mathematics and Statistics
Navier-Stokes equations
Tensors
title Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds
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