Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains
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Reinhard Farwig
Abstract
Let u be a weak solution of the Navier–Stokes equations in an exterior domain Ω ⊆ ℝ3 and a time interval [0,T[, 0 < T ≤ ∞, with initial value u0 and external force f = div F. We address the problem to find the optimal (weakest possible) initial value condition in order to obtain a strong solution u ∈ Ls(0,T; Lq(Ω)) in some time interval [0,T[, 0 < T < ∞, where s,q with 3 < q < ∞ and 2/s + 3/q = 1 are so-called Serrin exponents. Our main result states, for Serrin exponents s,q with 3 < q ≤ 8, a smallness condition on ∫0T || e-ντAu0 ||qsdτ to imply existence of a strong solution u ∈ Ls(0,T; Lq(Ω)); here A denotes the Stokes operator. Moreover, when 3 < q < ∞, we will prove the necessity of the condition ∫0∞ || e-ντAu0||qsdτ < ∞ to get a strong solution u on [0,T[, 0 < T ≤ ∞.
© by Oldenbourg Wissenschaftsverlag, München, Germany
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- Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains
- Hypergeometric summation representations of the Stieltjes constants
- Meromorphic functions whose certain differential polynomials share a small function with finite weight
- Convolutions of slanted half-plane harmonic mappings
- Meromorphic functions that share one small function DM with their first derivative
- A product theorem for the Euler and the Natarajan methods of summability
- Hermite–Hadamard type integral inequalities for geometric-arithmetically s-convex functions
Articles in the same Issue
- Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains
- Hypergeometric summation representations of the Stieltjes constants
- Meromorphic functions whose certain differential polynomials share a small function with finite weight
- Convolutions of slanted half-plane harmonic mappings
- Meromorphic functions that share one small function DM with their first derivative
- A product theorem for the Euler and the Natarajan methods of summability
- Hermite–Hadamard type integral inequalities for geometric-arithmetically s-convex functions
