Viktor Grigoryan | Searcharxiv

arXiv2017

Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D

In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Grünrock's result in 3D, we take the data in the Fourier-Lebesgue spaces $Ĥ_s^r$, which coincide with the Sobolev spaces of the same regularity for $r=2$, but scale like lower regularity Sobolev spaces for $1 1+\frac{3}{2r}$, $1<r\leq 2$. On one end this recovers the sharp result on the Sobolev scale, $H^{\frac{7}{4}+}$, while on the other end establishes the $Ĥ_{\frac{5}{2}}^{1+}$ result, which scales like the Sobolev $H^{\frac{3}{2}+}$, thus, corresponding to a $\frac{1}{4}$ derivative improvement on the Sobolev scale.

math.AP↗

arXiv2013

Almost critical well-posedness for nonlinear wave equation with $Q_{μν}$ 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 $Q_{μν}$. The Cauchy problem for these equations is known to be ill-possed for data in the Sobolev space $H^s$ with $s<5/4$ for all the basic null-forms, except $Q_0$. However, the scaling analysis predicts local well-posedness all the way to the critical regularity of $s_c=1$. Following Grünrock's result for the quadratic derivative NLW, we consider initial data in the Fourier-Lebesgue spaces $Ĥ_s^r$, which coincide with the Sobolev spaces of the same regularity for $r=2$, but scale like lower regularity Sobolev spaces for $1 1+{1}{r}$, $1<r\leq 2$, which at one extreme coincides with $H^{{3}{2}+}$ 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 $Ĥ_{2+}^{1+}$. Using appropriate multiplicative properties of the solution spaces and relying on bilinear estimates for the $Q_{μν}$ forms, we prove almost critical local well-posedness for the Ward wave map problem as well.

math.AP↗