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 Qμν . The Cauchy problem for these equations is known to be ill-posed for data in the Sobolev space H with s ≤ 5/4 for all the basic null-forms, except Q0, thus leaving a gap to the critical regularity of sc = 1. Following Grünrock’s result for the quadratic derivative NLW in 3D, we consider initial data in the Fourier-Lebesgue spaces Ĥ 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 H 5 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 Ĥ 2 . Using appropriate multiplicative properties of the solution spaces, and relying on bilinear estimates for the Qμν forms, we prove almost critical local well-posedness for the Ward wave map problem as well.