Non-uniqueness and \(h\)-principle for Hölder-continuous weak solutions of the Euler equations. (English) Zbl 1372.35221
The incompressible Euler equations \(v_t+\operatorname{div}(v \otimes v) + \nabla p =0\), \(\operatorname{div}v=0\) are considered in the periodic setting \(\mathbb T^3= \mathbb R^3 / \mathbb Z^3\). A given initial data satisfies the Hölder condition with the Hölder exponent \(\theta \in (0,1)\). The authors prove that for \(\theta <1/5 - \varepsilon\) with sufficiently small \(\varepsilon>0\) there exist infinitely many admissible Hölder \(1/5 - \varepsilon\) weak solutions. A new set of stationary flows is introduced in order to show that a general form of the \(h\)-principle applies to Hölder-continuous weak solutions of the Euler equations.
Reviewer: Vladimir Mityushev (Kraków)
MSC:
| 35Q31 | Euler equations |
| 35D30 | Weak solutions to PDEs |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
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