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

Authors

Viktor Grigoryan (Department of Mathematics, Occidental College, Los Angeles, California, U.S.A.)

Andrea R. Nahmod (Department of Mathematics, University of Massachusetts, Amherst, Mass., U.S.A.)

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

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Published 1 August 2014