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# Mathematical Research Letters

## Volume 21 (2014)

### Number 2

### Almost critical well-posedness for nonlinear wave equations with $Q_{\mu \nu}$ null forms in 2D

Pages: 313 – 332

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n2.a9

#### Authors

#### Abstract

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 $s \leq 5/4$ for all the basic null forms, except $Q_0$, thus leaving a gap to the critical regularity of $s_c = 1$. Following Grünrock’s result for the quadratic derivative NLW in three dimensions, we consider initial data in the Fourier-Lebesgue spaces $\hat{H}^r_s$, which coincide with the Sobolev spaces of the same regularity for $r = 2$, but scale like lower regularity Sobolev spaces for $1 \lt r \lt 2$. Here we obtain local well-posedness for the range $s \gt \frac{3}{2r} + \frac{1}{2} , 1 \lt r \leq 2$, which at one extreme coincides with $H^{\frac{5}{4}+}$ 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 $\hat{H}^1_2 {}+$. 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.

#### 2010 Mathematics Subject Classification

35L70