{
  "id": "1307.6194",
  "version": "v1",
  "published": "2013-07-23T19:09:27.000Z",
  "updated": "2013-07-23T19:09:27.000Z",
  "title": "Almost critical well-posedness for nonlinear wave equation with $Q_{μν}$ null forms in 2D",
  "authors": [
    "Viktor Grigoryan",
    "Andrea R. Nahmod"
  ],
  "categories": [
    "math.AP"
  ],
  "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-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 Gr\\\"{u}nrock's result for the quadratic derivative NLW, we consider initial data in the Fourier-Lebesgue spaces $\\^{H}_s^r$, 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 $s>1+{1}{r}$, $1<r\\leq 2$, 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 $\\^{H}_{2+}^{1+}$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-23T19:09:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L70"
    ],
    "keywords": [
      "nonlinear wave equation",
      "null forms",
      "critical well-posedness",
      "fourier-lebesgue space",
      "extreme establishes local well-posedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6194G"
    }
  }
}