## Abstract

We consider the stationary Navier–Stokes equations in a bounded domain *Ω* in **R**
^{n} with smooth connected boundary, where *n* = 2, 3 or 4. In case that *n* = 3 or 4, existence of very weak solutions in *L*
^{n} (*Ω*) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete *L*
^{q}-regularity results on very weak solutions in *L*
^{n} (*Ω*). If *n* = 2, then similar results are also proved for very weak solutions in \({L^{q_0} (\Omega )}\) with any *q*
_{0} > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results.

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Communicated by C. Le Bris

An erratum to this article can be found at http://dx.doi.org/10.1007/s00205-010-0310-1

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Kim, H. Existence and Regularity of Very Weak Solutions of the Stationary Navier–Stokes Equations.
*Arch Rational Mech Anal* **193**, 117–152 (2009). https://doi.org/10.1007/s00205-008-0168-7

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DOI: https://doi.org/10.1007/s00205-008-0168-7