On the Schrödinger-Debye System in Compact Riemannian Manifolds

Abstract

We consider the initial value problem (IVP) associated to the Schrödinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M$ and prove local well-posedness result for given data $(u_0,v_0)\in H^s(M)\times (H^s(M)\cap L^{\infty }(M))$ whenever $s>\frac{d}{2}-\frac{1}{2}$, $d\ge 2$. For $d=2$, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the $H^1$-solution and use it to prove the global well-posedness result in this space.

31 pages