# Well-Posedness and Finite-Time Blowup for the Zakharov System on Two-Dimensional Torus

# Well-Posedness and Finite-Time Blowup for the Zakharov System on Two-Dimensional Torus

We consider the Zakharov system on two-dimensional torus. First, we show the local well-posedness of the Cauchy problem in the energy space by a standard iteration argument using the $X^{s,b}$ norms. Our result does not depend on the period of torus. Conservation laws and a sharp Gagliardo-Nirenberg inequality imply an a priori bound of solutions, which enables us to extend the local-in-time solution to a global one if its L2 norm is less than that of the ground state solution of the cubic NLS on R2. We then show that the L2 norm of the ground state is actually the threshold for global solvability, namely, that there exists a finite-time blow-up solution to the Zakharov system on 2d torus with the L2 norm greater than but arbitrarily close to that of the ground state. This is joint work with Masaya Maeda (Tohoku University, Japan).