ALMOST CRITICAL WELL-POSEDNESS FOR NONLINEAR WAVE EQUATIONS WITH Qμν NULL FORMS IN 2D

In this paper, we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities Q mu nu. The Cauchy problem for these equations is known to be ill-posed for data in the Sobolev space H-s with s = 5/4 for all the basic null forms, except Q(0), thus leaving a gap to the critical regularity of s(c) = 1. Following Grunrock's result for the quadratic derivative NLW in three dimensions, we consider initial data in the Fourier-Lebesgue spaces (H) over cap (r)(s), which coincide with the Sobolev spaces of the same regularity for r = 2, but scale like lower regularity Sobolev spaces for 1 < r < 2. Here we obtain local well-posedness for the range s > 3/2r + 1/2, 1 < r <= 2, which at one extreme coincides with H5/4+ optimal Sobolev space result, while at the other extreme establishes local well-posedness for the model null-form problem for the almost critical Fourier-Lebesgue space <(H)over cap>(1+)(2). Using appropriate multiplicative properties of the solution spaces, and relying on bilinear estimates for the Q mu nu forms, we prove almost critical local well-posedness for the Ward wave map problem as well.