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Anomalous dissipation for $$1/5$$-Hölder Euler flows. (English) Zbl 1330.35303
The authors consider the incompressible Euler equations, and study the celebrated Onsager conjecture. The conjecture states that (i) if the solution has a Hölder exponent $$\theta$$ satisfying $$\theta >1/3$$, then the energy integral of the solution is conserved, and (ii) if it satisfies $$\theta >1/3$$, then the energy is not necessarily conserved. The authors study the latter statement, and discuss about the Hölder exponent. They consider the case $$\theta <1/5$$ on a three-dimensional torus $$\mathbf{T}\times \mathbf{T}\times\mathbf{T}$$, which means to consider periodic soluitons. They prove that for any positive smooth function $$e(t)$$ there exists a solution of the incompressible Euler equations whose energy integral is equal to this function. This means that the Onsager conjecture for this case is true.
They use iteration for $$q=1,2,3,\cdots$$ At each step $$q$$, they construct a triple $$(v_q ,p_q ,R_q )$$ consisting of velocity $$v_q$$, pressure $$p_q$$, and the Reynolds stress $$R_q$$ which satisfy the following Euler Reynolds system: $\partial_t v_q +\roman{div}\!\;(v_q \otimes v_q )+\nabla p_q =\roman{div}\!\;R_q ,\;\roman{div}\!\;v_q =0 .$ The iteration proceeds so that $$(v_q ,p_q , R_q )\rightarrow (v,p,0)$$ in the Hölder space when $$q\rightarrow \infty$$, and $$\int|v_q |^2 dx$$ converges to a given function $$e(t)$$. Technical parts are collected in the appendix, which makes this article more accessible.

##### MSC:
 35Q31 Euler equations 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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