By Gregory Seregin
The lecture notes during this booklet are in line with the TCC (Taught direction Centre for graduates) path given by way of the writer in Trinity phrases of 2009 2011 on the Mathematical Institute of Oxford college. It includes roughly an user-friendly creation to the mathematical idea of the Navier Stokes equations in addition to the fashionable regularity conception for them. The latter is built via the classical PDE's concept within the variety that's rather regular for St Petersburg's mathematical tuition of the Navier Stokes equations. the worldwide precise solvability (well-posedness) of preliminary boundary price difficulties for the Navier Stokes equations is in truth one of many seven Millennium difficulties said by way of the Clay Mathematical Institute in 2000. It has now not been solved but. although, a deep connection among regularity and well-posedness is understood and will be used to assault the above demanding challenge. this kind of method isn't very good provided within the smooth books at the mathematical thought of the Navier Stokes equations. including advent chapters, the lecture notes can be a self-contained account at the subject from the very simple stuff to the state-of-art within the field.
Readership: Undergraduate and graduate scholars in differential equtions and fluid mechanics.
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Extra info for Lecture Notes on Regularity Theory for the Navier-Stokes Equations
10, there exists p ∈ L2 (Ω) such that l(v) = pdiv vdx Ω ◦ for any v ∈ L12 (Ω). Therefore, u = ∇p and thus p ∈ G(Ω) and ◦ (J (Ω))⊥ ⊆ G(Ω). 10. Consider a sequence of domains Ωj with the properties: Ωj ⊂ Ωj+1 and Ω= ∞ Ωj , j=1 where Ωj is a bounded Lipschitz domain. Since v ∈ L2 (Ω) ⇒ v ∈ L2 (Ωj ), we can state that, for any j, v = u(j) + ∇p(j) , where ◦ u(j) ∈ J (Ωj ), p(j) ∈ W21 (Ωj ). We know that p(j) is defined up to a constant, which can be fixed by the condition p(j) dx = 0, B∗ where B∗ is a fixed ball belonging to Ω1 .
There is a wide class of domains for which the above spaces coincide. 8. Let 1 < m < ∞ and let Ω be Rn , or Rn+ , or a bounded domain with Lipschitz boundary. Then ◦ 1 1 J m (Ω) = Jˆm (Ω). Proof See next section. 9. Let Ω = Rn or Rn+ or be a bounded Lipschitz domain. ◦ Assume that 1 < s < ∞. Let, further, l : L1s (Ω) → R be a linear functional having the following properties: |l(v)| ≤ c ∇v ◦ s,Ω for any v ∈ L1s (Ω) and l(v) = 0 for any v ∈ Jˆ1s (Ω). Then there exists a function p ∈ Ls′ (Ω), s′ = s/(s − 1), such that l(v) = pdiv vdx Ω ◦ for any v ∈ L1s (Ω).
11. (Ladyzhenskaya) For any domain Ω ∈ Rn , ◦ L2 (Ω) := J (Ω) ⊕ G(Ω). Proof Obviously, our statement is equivalent to the following identity ◦ G(Ω) = (J (Ω))⊥ . Step 1 Let Ω be a bounded Lipschitz domain. It is easy to see that ◦ G(Ω) ⊆ (J (Ω))⊥ , since v · ∇pdx = 0 Ω ∞ for any p ∈ G(Ω) and for any v ∈ C0,0 (Ω). , u ∈ L2 (Ω) and u · vdx = 0 Ω for any v ∈ ∞ C0,0 (Ω). By Poincar´e inequality, |u|2 u · vdx ≤ l(v) = Ω 2,Ω |v|2 Ω Ω ≤ c(Ω) u 1 2 ∇v 2,Ω 1 2 ≤ page 26 August 27, 2014 14:25 LectureNotesonLocalRegularity LectureNotes Preliminaries ◦ ◦ 27 ◦ for any v ∈ L12 (Ω).