Geodesic
Examples of Equations of Navier--Stokes Type Not Strongly Solvable in the Large
Matematičeskie zametki, Tome 89 (2011) no. 5, pp. 771-779

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In the space of functions with values in Hilbert space, we consider the Cauchy problem $u'_t+Au+B(u,u)=f(t)$, $u(0)=0$, $0\le t\le T$. We construct examples of a self-adjoint operator $A\ge E$ and a bilinear transformation $B$ satisfying the condition $\langle B(u,v),v\rangle=0$ such that the Cauchy problem is not strongly solvable.
Keywords: Navier–Stokes equation, self-adjoint operator, bilinear transformation, Cauchy problem, Hilbert space, Sobolev class. 
@article{MZM_2011_89_5_a10,
     author = {M. Otelbaev},
     title = {Examples of {Equations} of {Navier--Stokes} {Type} {Not} {Strongly} {Solvable} in the {Large}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {771--779},
     publisher = {mathdoc},
     volume = {89},
     number = {5},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_5_a10/}
}
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M. Otelbaev. Examples of Equations of Navier--Stokes Type Not Strongly Solvable in the Large. Matematičeskie zametki, Tome 89 (2011) no. 5, pp. 771-779. http://geodesic.mathdoc.fr/item/MZM_2011_89_5_a10/