Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid.

*(English)*Zbl 0874.93055The aim of this paper is to construct a generalized Lyapunov function and a generalized integral for an ideal incompressibe fluid, namely the case of a fluid moving on a standard 2-dimensional torus \(\mathbb{T}^2\).

The configuration space \({\mathcal D}B_s\) consists of all diffeomorphisms \(g\mathbb{T}^2\to \mathbb{T}^2\) such that \(g(x)-x\) belongs to the Besov space \(B_s\) and the vorticity belongs to \(B_{s-t}\). The phase space is \(X_s={\mathcal D}B_s\times B_{s-1}\) with Besov space topology. The vorticity of the fluid flow is defined by \(\omega(y,t)= \text{rot } v(y,t)\), where \(v(y,t)= V(g_t^{-1}(y),t)\) is the Eulerian velocity and \(V(x,t)= \partial g_t(x)/\partial t\) is the Lagrangian velocity. The motion equations are: \[ (d/dt)(g,\omega)= (\text{rot}^{-1}(\omega\circ g^{-1})\circ g,0). \tag{\(*\)} \] For \(s>3\), the author constructs a topological vector space \(V^{2s- 2}\), semi-ordered by a convex cone \(V_+^{2s-2}\subset V^{2s-2}\), and the continuous mapping \(L:X_s\to V^{2s-2}\), \(L(g,\omega)= [(\text{rot } W_g,\omega)_{ml}]\in V^{2s-2}\), where \([(\text{rot } W_g,\omega)_{ml}]\) is the equivalence class of certain measure \((\text{rot } W_g,\omega)_{ml}\) associated to each element \((g,\omega)\in X_s\). This mapping satisfies \((d/dt)L(g,\omega)\in V_+^{2s-2}\) for each solution \((g(t),\omega)\) of \((*)\) and it is strictly growing in an invariant, dense and open set \(Y_s\); therefore \(L\) is a generalized Lyapunov function.

Likewise the author constructs a generalized integral \(M_\omega\) of the motion equation \((*)\) defined on the fiber \({\mathcal D}B_s\times \{\omega\}\) with values in a topological vector space \(W_\omega^{2s-2}\), by \(M_\omega(g)= \pi L(g,\omega)\), where \(\pi:V^{2s-2}\to W_\omega^{2s-2}\) is the natural projection. This is a continuous function which is constant along every trajectory.

An important merit of this work is the use of the notions of paradifferential calculus (a branch of microlocal analysis created by Bony, Meyer, Alinhac and others), as main technical tools (paraproduct and paracomposition of finitely smooth functions). Moreover, we remark the use of the complete linearization formula obtained by S. Alinhac: \[ u\circ g=T_{du\circ g}g+g^*u+R(u,g), \] where the first term bears the singularities of \(g\), and the second the singularities of \(u\). So, these two kinds of singularities, mixed in the usual composition, are separated by the paraproduct and paracomposition. The author obtains a precise result concerning propagation of singularities, using the generalized Lyapunov function. Finally, the author indicates a constructive way to obtain many scalar valued Lyapunov functions and new integrals for the fluid.

The configuration space \({\mathcal D}B_s\) consists of all diffeomorphisms \(g\mathbb{T}^2\to \mathbb{T}^2\) such that \(g(x)-x\) belongs to the Besov space \(B_s\) and the vorticity belongs to \(B_{s-t}\). The phase space is \(X_s={\mathcal D}B_s\times B_{s-1}\) with Besov space topology. The vorticity of the fluid flow is defined by \(\omega(y,t)= \text{rot } v(y,t)\), where \(v(y,t)= V(g_t^{-1}(y),t)\) is the Eulerian velocity and \(V(x,t)= \partial g_t(x)/\partial t\) is the Lagrangian velocity. The motion equations are: \[ (d/dt)(g,\omega)= (\text{rot}^{-1}(\omega\circ g^{-1})\circ g,0). \tag{\(*\)} \] For \(s>3\), the author constructs a topological vector space \(V^{2s- 2}\), semi-ordered by a convex cone \(V_+^{2s-2}\subset V^{2s-2}\), and the continuous mapping \(L:X_s\to V^{2s-2}\), \(L(g,\omega)= [(\text{rot } W_g,\omega)_{ml}]\in V^{2s-2}\), where \([(\text{rot } W_g,\omega)_{ml}]\) is the equivalence class of certain measure \((\text{rot } W_g,\omega)_{ml}\) associated to each element \((g,\omega)\in X_s\). This mapping satisfies \((d/dt)L(g,\omega)\in V_+^{2s-2}\) for each solution \((g(t),\omega)\) of \((*)\) and it is strictly growing in an invariant, dense and open set \(Y_s\); therefore \(L\) is a generalized Lyapunov function.

Likewise the author constructs a generalized integral \(M_\omega\) of the motion equation \((*)\) defined on the fiber \({\mathcal D}B_s\times \{\omega\}\) with values in a topological vector space \(W_\omega^{2s-2}\), by \(M_\omega(g)= \pi L(g,\omega)\), where \(\pi:V^{2s-2}\to W_\omega^{2s-2}\) is the natural projection. This is a continuous function which is constant along every trajectory.

An important merit of this work is the use of the notions of paradifferential calculus (a branch of microlocal analysis created by Bony, Meyer, Alinhac and others), as main technical tools (paraproduct and paracomposition of finitely smooth functions). Moreover, we remark the use of the complete linearization formula obtained by S. Alinhac: \[ u\circ g=T_{du\circ g}g+g^*u+R(u,g), \] where the first term bears the singularities of \(g\), and the second the singularities of \(u\). So, these two kinds of singularities, mixed in the usual composition, are separated by the paraproduct and paracomposition. The author obtains a precise result concerning propagation of singularities, using the generalized Lyapunov function. Finally, the author indicates a constructive way to obtain many scalar valued Lyapunov functions and new integrals for the fluid.

Reviewer: M.Ivanovici (Craiova)

##### MSC:

93C20 | Control/observation systems governed by partial differential equations |

93D30 | Lyapunov and storage functions |

76B47 | Vortex flows for incompressible inviscid fluids |