Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations
In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.
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Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations Pavlo O. Kasyanov, Luisa Toscano and Nina V. Zadoianchuk
Abstract In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.
13.1 Introduction Let Ω ⊆ R3 be a bounded open set with sufficiently smooth boundary ∂Ω and 0 < T < +∞. We consider the incompressible Navier-Stokes equations ⎧ ⎨ yt + (y · ∇)y = νy − ∇p + f in Q = Ω × (0, T ), div y = 0 in Q, ⎩ y = 0 on ∂Ω × (0, T ), y(x, 0) = y0 (x) in Ω,
(13.1)
where ν > 0 is a constant. We define the usual function spaces V = {u ∈ (C0∞ (Ω))3 : div u = 0}, H = closure of V in (L 2 (Ω))3 , V = {u ∈ (H01 (Ω))3 : div u = 0}. P. O. Kasyanov (B) Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave., 37, build, 35, Kyiv 03056, Ukraine e-mail: [email protected] L. Toscano Department of Mathematics and Applications R. Caccioppoli, University of Naples “Federico II”, via Claudio 21, 80125 Naples, Italy e-mail: [email protected] N. V. Zadoianchuk Department of Computational Mathematics, Taras Shevchenko National University of Kyiv, Volodimirska Street 64, Kyiv 03601, Ukraine e-mail: [email protected] M. Z. Zgurovsky and V. A. Sadovnichiy (eds.), Continuous and Distributed Systems, Solid Mechanics and Its Applications 211, DOI: 10.1007/978-3-319-03146-0_13, © Springer International Publishing Switzerland 2014
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We denote by V ∗ the dual space of V . The spaces H and V are separable Hilbert spaces and V ⊂ H ⊂ V ∗ with dense and compact embedding when H is identified with its dual H ∗ . Let (·, ·), · H and ((·, ·)), · V be the inner product and the norm in H and V , respectively, and let ·, · be the pairing between V and V ∗ . For u, v, w ∈ V , the equality 3
b(u, v, w) =
ui
Ω i,j=1
∂vj wj dx ∂xi
defines a trilinear continuous form on V with b(u, v, v) = 0 when u ∈ V and v ∈ (H01 (Ω))3 . For u, v ∈ V , let B(u, v) be the element of V ∗ defined by B(u, v), w = b(u, v, w) for all w ∈ V . We say that the function y is a weak solution of Problem (13.1) on [0, T ], if 1 ∗ y ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ), dy dt ∈ L (0, T ; V ), if d (y, v) + ν((y, v)) + b(y, y, v) = f , v for all v ∈ V , dt
(13.2)
in the sense of distributions on (0, T ), and if y satisfies the energy inequality V (y)(t) ≤ V (y)(s) for all t ∈ [s, T ],
(13.3)
for a.e. s ∈ (0, T ) and for s = 0, where 1 V (y)(t) : = y(t) 2H + ν 2
t
t y(τ ) 2V dτ
0
−
f (τ ), y(τ ) dτ.
(13.4)
0
This class of solutions is called Leray–Hopf or physical one. If f ∈ L 2 (0, T ; V ∗ ), and 4 ∗ 3 if y satisfies (13.2), then y ∈ C([0, T ]; Hw ), dy dt ∈ L (0, T ; V ), where Hw denotes the space H endowed with the weak topology. In particular, the initial condition y(0) = y0 makes sense for any y0 ∈ H. Let A : V → V ∗ be the linear operator associated to the bilinear form ((u, v)) =
Au, v . Then A is an isomorphism from D(A) onto H with D(A) = (H 2 (Ω))3 ∩ V
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