This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters $h$ and $\Delta t$. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order $h^2$ and $\Delta t^2$. Using a discrete...

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using...

A uniqueness criterion is given for the weak solution of the Navier-Stokes equations in the stationary case. Precisely, it is proved that, for a generic known term, there exists one and only one solution such that the mechanical power of the corresponding flow is maximum and that this maximum is "stable" in an appropriate sense.

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_i$ of the fluids and their velocity fields $u^{\left(i\right)}$ are prescribed at infinity: ${\rho }_i{|}_{\infty }={\rho }_{i\infty }>0$, $u^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_i\equiv {\rho }_{i\infty }$, $u^{\left(i\right)}\equiv 0$, $i=1,2$.

We establish a uniqueness result for an overdetermined boundary value problem. We also raise a new question.

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with bounded divergence....

It is proved that there can exist at most one solution of the homogeneous Dirichlet problem for the stationary Navier-Stokes equations in 3-dimensional space which is approximable by a given consistent and regular approximation scheme.

In questa Nota si fornisce un teorema di unicità per soluzioni regolari delle equazioni di Navier-Stokes in domini esterni. Tale teorema non richiede che le velocità tendano ad un prefissato limite all'infinito, mentre il gradiente di pressione è supposto essere di $q$-ma potenza sommabile nel cilindro spazio-temporale $(q\in (1,\mathrm{\infty }))$. Questo risultato non può essere ulteriormente generalizzato al caso $q=\mathrm{\infty }$, a causa di noti controesempi.

Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k\left(t\right)$ with respect to some point $p\in M$, where $t=d(·,p)$ is the Riemannian distance on $M$ to $p$, $k\left(t\right)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty$, satisfy $$\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}+\cdots +\frac{1}{{\mu }_n}\ge \frac{l^{n+2}}{(n+2){\int }_0^l{f}^{n-1}\left(t\right)\mathrm{d}t},$$ where $f\left(t\right)$ is the solution to $$\left\{\begin{array}{c}{f}^{\text{'}\text{'}}\left(t\right)+k\left(t\right)f\left(t\right)=0\text{on}(0,\infty ),\hfill \\ f\left(0\right)=0,f^{\text{'}}\left(0\right)=1.\hfill \end{array}\right.$$