Local and global solutions on arcs for the Ericksen-Leslie problem in \(\mathbb{R}^N\) (Q6648864)

The authors prove the existence and decay estimates of solutions to the Ericksen-Leslie model written as: \((\partial _{t}-\Delta)u+\nabla p+u\cdot \nabla u=-\operatorname{Div}(\nabla v\odot \nabla v)\), \(\operatorname{div}u=0\), \((\partial _{t}-\Delta)v+u\cdot \nabla v=\left\vert \nabla v\right\vert ^{2}v\), \(\left\vert v\right\vert =1\), in \(\mathbb{R}_{+}\times \mathbb{R}^{N}\), with \((\nabla w\odot \nabla z)_{jk}=\partial _{j}w\cdot \partial _{k}v\), \(j, k=1, \ldots, N\). \N\NConsidering two fixed vectors \(\eta, \omega \in \mathbb{S}^{N-1}\), with \(\eta \perp \omega\), and assuming that the initial data \(v_{0}\) is on the arc joining \(\eta\) and \(\omega\), that is \(v_{0}(x)=\cos d_{0}(x)\eta +s\operatorname{ind}_{0}(x)\omega\) for a.e. \(x\in \mathbb{R}^{N}\), the authors expect that \(v\) remains on the same arc, that is, \(v(t, x)=\cos d(t, x)\eta +s\operatorname{ind}(t, x)\omega\) for a.e. \((t, x)\in \mathbb{R}_{+}\times \mathbb{R}^{N}\) and they rewrite the above problem as the reduced one: \((\partial _{t}-\Delta)u+\nabla p+u\cdot \nabla u=-\operatorname{Div}(\nabla d\odot \nabla d)\), \(\operatorname{div}u=0\), \((\partial _{t}-\Delta)d+u\cdot \nabla d=0\), in \(\mathbb{R}_{+}\times \mathbb{R}^{N}\), with the initial conditions \(u(0)=u_{0}\), \(d(0)=d_{0}\). They prove that if \(N\geq 3\) and \(s>N/2-1\), for any \(R>0\), there exists \(T(R)>0\) so that for any initial data \(u_{0}\in J_{2}(\mathbb{R}^{N})\cap H^{s}(\mathbb{R}^{N}; \mathbb{R}^{N})\), \(d_{0}\in L^{\infty }(\mathbb{R}^{N})\), such that \(\nabla d_{0}\in H^{s}(\mathbb{R}^{N}; \mathbb{R}^{N})\), and \(\left\Vert u_{0}\right\Vert _{H^{s}(\mathbb{R}^{N})}+\left\Vert d_{0}\right\Vert _{L^{\infty }(\mathbb{R}^{N})}+\left\Vert \nabla d_{0}\right\Vert _{H^{s}(\mathbb{R}^{N})}\leq R\), there exists a solution \((u, p, d)\) in appropriate spaces to the reduced system in \((0, T)\), that is unique up to additive functions \(\rho (t)\) on the pressure \(p\). Here \(J_{2}(\mathbb{R}^{N})=\{w\in L^{2}(\mathbb{R}^{N}; \mathbb{R}^{N})\mid \left\langle w, \nabla \varphi \right\rangle =0\), \(\forall \lbrack \varphi ]\in \widehat{H}^{1}(\mathbb{R} ^{N})\}\). If the initial data are small enough, this solution is global in time and the authors prove a decay estimate with respect to time. \N\NComing back to the Ericksen-Leslie system, the authors consider the case where \(N=3\) and \(s\in \mathbb{N}\) and \(s>N/2-1\). If the initial data satisfy \(u_{0}\in J_{2}(\mathbb{R}^{N})\cap H^{s}(\mathbb{R}^{N}; \mathbb{R}^{N})\), \(v_{0}: \mathbb{R}^{N}\rightarrow \mathbb{S}^{N-1}\), belongs to \(\operatorname{Span}\left\{ \eta, \omega \right\}\) with \(\eta \perp \omega\) and \(\nabla v_{0}\in H^{s}(\mathbb{R}^{N}; \mathbb{R}^{N})\), and \(\left\Vert u_{0}\right\Vert _{H^{s}(\mathbb{R}^{N})}+\left\Vert v_{0}-\eta \right\Vert _{L^{\infty }(\mathbb{R} ^{N})}+\left\Vert \nabla v_{0}\right\Vert _{H^{s}(\mathbb{R}^{N})}\leq R\), there exists a solution \((u, p, v)\) in appropriate spaces, unique up to additive functions \(\rho (t)\) on the pressure \(p\), for the Ericksen-Leslie system in \((0, T)\). If \(N=3\) and \(s\in (1/2, 1)\), and the initial data satisfy the preceding hypotheses and the sum of the three preceding norms is small enough, the solution is global in time and satisfies some decay estimate with respect to time. For the proof, the authors introduce the linear system: \((\partial _{t}-\Delta)u+\nabla p=f\), \(\operatorname{div}u=0\), \((\partial _{t}-\Delta)d=g\), in \(\mathbb{R}_{+}\times \mathbb{R}^{N}\), with the initial conditions \(u(0)=u_{0}\), \(d(0)=d_{0}\), the Leray projection, and the heat and Stokes semigroups. They prove estimates on functions involving the heat and Stokes semigroups.