Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport. (A compactness result for transport equations and application to the calculus of the limit of the principal eigenvalue of a transport operator). (French) Zbl 0591.45007

Golse, François; Perthame, Benoît; Sentis, Rémi

C. R. Acad. Sci., Paris, Sér. I 301, 341-344 (1985).

Let X be a convex, bounded, open set of \({\mathbb{R}}^ N\), with a regular boundary \(\partial X\), V a compact set of \({\mathbb{R}}^ N\) such that: if \(N=1\), then \(V=[-1,1]\), if \(N>1\), then V is rotationally invariant and \(0\not\in V\). Let \(\Gamma =\{x\in \partial X,v\in V| n_ x\cdot v<0\}\), where \(n_ x\) is the external normal to \(\partial X\) at x. Let, moreover, \(A^{\epsilon}\) be the linear transport operator \[ A^{\epsilon}u=-\frac{1}{\epsilon}v\cdot \frac{\partial u}{\partial x}+\frac{1}{\epsilon^ 2}Qu\quad (x\in X,\quad v\in V,\quad \epsilon >0), \] defined on the domain \(D(A^{\epsilon})=\{u(x,v)\in L^ 2(X\times V):\| u\|_{L^ 2(X\times V)}\leq C,\| v\cdot \partial u/\partial x\|\) \(_{L^ 2(X\times V)}\leq C\), \(u=0\) for (x,v)\(\in \Gamma \}\), where \((Qu)(x,v)=\int_{v}\sigma_ 1(x,v,w)u(x,w)dw-\sigma (x,v)u(x,v)\) with \(\sigma (x,v)=\int_{v}\sigma_ 1(x,v,w)dw\) is the collision operator, here restricted to pure scattering (although extension to more general problems should be not too difficult). Let, finally, \(\omega_{\epsilon}\) be the fundamental eigenvalue of \(A^{\epsilon}\), \(\phi_{\epsilon}\in L^ 2(X\times V)\) the positive eigenfunction belonging to it and \(A\bar u=\sum_{ij}\frac{\partial}{\partial x_ i}a_{ij}\frac{\partial \bar u}{\partial x_ j}\), \(D(A)=H^ 2(X)\cap H^ 1_ 0(X)\) the classical (non-absorption) diffusion operator defined in \(L^ 2(X)\), with \(\omega\) as its fundamental eigenvalue and \(\phi \in L^ 2(X)\) as the corresponding positive eigenfunction. The authors announce (with short sketches of the proofs) the following result: the family \(\{{\tilde \phi}_{\epsilon}\}\), with \({\tilde \phi}_{\epsilon}(x)=\int_{v}\phi_{\epsilon}(x,v)dv\) is relatively compact in \(L^ 2(X)\) and a subsequence of it converges to \({\bar \phi}\) in \(L^ 2(X)\). Moreover, \(\omega_{\epsilon}\to \omega\) and \(\phi_{\epsilon}\) tends to a limit function in \(L^ 2(X\times V)\) which, projected on \(L^ 2(X)\), also coincides with \({\bar \phi}\).

Reviewer: B.Montagnini

MSC:

45K05	Integro-partial differential equations
45C05	Eigenvalue problems for integral equations
82C70	Transport processes in time-dependent statistical mechanics

Keywords:

compactness; linear transport operator; collision operator; scattering; fundamental eigenvalue; positive eigenfunction; diffusion operator