On a conjecture due to J. L. Lions. (English) Zbl 0844.35082

Fernández-Cara, Enrique; Real, José

Nonlinear Anal., Theory Methods Appl. 21, No. 11, 835-847 (1993).

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^3\) and \(\omega\subset \Omega\). \(( v, p)\) is the solution in the sense of distributions to the initial-boundary value problem for Navier-Stokes equations \[ v_t+ ( v\cdot \nabla) v -\nu\Delta v+ \nabla p= X_\omega f,\;\nabla\cdot v= 0\quad\text{in }\Omega\times (0, T),\tag{1} \]
\[ v= 0\quad\text{on} \quad \partial\Omega\times (0, T),\quad v(x, 0)= 0, \] where \( f\in L_2(\omega\times (0, T))\). \(X_\omega\) is the characteristic function on the set \(\omega\). \(u( f)\) denotes the set of functions \( v\) such that \(( v, p)\) is a weak solution of the problem (1). It is proved that the subspace of \(H(\Omega)\) which is spanned by \((\cup_f u( f))\cap H\) is dense in \(H\). For the two-dimensional case this result has been obtained by the authors [in ‘On a conjecture due to J. L. Lions concerning weak controllability of Navier-Stokes flows’, Proc. Equadiff ’91 Conference, Barcelona, C. Perello (ed.) et al., 126-135 (World Scientific 1991)].

Reviewer: I.Sh.Mogilevskij (Ferrara)

MSC:

35Q30	Navier-Stokes equations
35D05	Existence of generalized solutions of PDE (MSC2000)
93B05	Controllability

Keywords:

weak controllability of Navier-Stokes flows

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