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Existence of weak solutions for the incompressible Euler equations. (English) Zbl 1228.35172
Summary: Using a recent result of C. De Lellis and L. Székelyhidi jun. [Arch. Ration. Mech. Anal. 195, No. 1, 225–260 (2010; Zbl 1192.35138)] we show that, in the case of periodic boundary conditions and for arbitrary space dimension \(d \geq 2\), there exist infinitely many global weak solutions to the incompressible Euler equations with initial data \(v_0\), where \(v_0\) may be any solenoidal \(L^2\)-vector field. In addition, the energy of these solutions is bounded in time.

MSC:
35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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[2] De Lellis, Camillo; Székelyhidi, László, On admissibility criteria for weak solutions of the Euler equations, Arch. ration. mech. anal., 195, 1, 225-260, (2010) · Zbl 1192.35138
[3] Leray, Jean, Sur le mouvement dʼun liquide visqueux emplissant lʼespace, Acta math., 63, 1, 193-248, (1934) · JFM 60.0726.05
[4] Székelyhidi, László; Wiedemann, Emil, Generalised Young measures generated by ideal incompressible fluid flows, (2011), preprint: · Zbl 1256.35072
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