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.


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