Jean Van Schaftingen

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The function spaces Dk(R) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp = n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div–curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539–543 (see also J. Van(More)
Article history: Received 4 September 2015 Accepted after revision 8 October 2015 Available online 6 November 2015 Presented by Haïm Brézis As a consequence of inequalities due to Bourgain–Brézis, we obtain local-in-time wellposedness for the two-dimensional Navier–Stokes equation with velocity bounded in spacetime and initial vorticity in bounded(More)
We study various questions related to the best constants in the following inequalities established in [1, 2, 3]; ̨̨̨Z Γ ~ φ · ~t ̨̨̨ ≤ Cn‖∇φ‖Ln |Γ| , and ̨̨̨Z Rn ~ φ · ~ μ ̨̨̨ ≤ Cn‖∇φ‖Ln‖~ μ‖ , where Γ is a closed curve in Rn, ~ φ ∈ C∞ c (Rn; Rn) and ~ μ is a bounded measure on Rn with values into Rn such that div ~ μ = 0. In 2d the answers are rather(More)
M 〈f, φ〉dV ∣∣∣∣ ≤ C‖f‖L1(dV )‖∇φ‖Lm(dV ). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces. © 2017 Elsevier Inc. All rights reserved. * Corresponding author. E-mail addresses: (S. Chanillo), (J. Van Schaftingen), (P.-L.(More)