Anomalous dissipation for \(1/5\)-Hölder Euler flows. (English) Zbl 1330.35303

Buckmaster, Tristan; De Lellis, Camillo; Isett, Philip; Székelyhidi, László jun.

Ann. Math. (2) 182, No. 1, 127-172 (2015).

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.

Reviewer: Keisuke Uchikoshi (Yokosuka)

MSC:

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

Keywords:

Onsager’s conjecture; anomalous dissipation; turbulant flows

References:

[1]	T. Buckmaster, ”Onsager’s conjecture almost everywhere in time,” Comm. Math. Phys., vol. 333, p. 24, 2015. · Zbl 1308.35184
[2]	A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy, ”Energy conservation and Onsager’s conjecture for the Euler equations,” Nonlinearity, vol. 21, iss. 6, pp. 1233-1252, 2008. · Zbl 1138.76020
[3]	P. Constantin, W. E, and E. S. Titi, ”Onsager’s conjecture on the energy conservation for solutions of Euler’s equation,” Comm. Math. Phys., vol. 165, iss. 1, pp. 207-209, 1994. · Zbl 0818.35085
[4]	S. Conti, C. De Lellis, and L. relax Székelyhidi Jr., ”\(h\)-principle and rigidity for \(C^{1,\alpha}\) isometric embeddings,” in Nonlinear Partial Differential Equations, New York: Springer-Verlag, 2012, vol. 7, pp. 83-116. · Zbl 1255.53038
[5]	R. Courant, K. Friedrichs, and H. Lewy, ”On the partial difference equations of mathematical physics,” IBM J. Res. Develop., vol. 11, pp. 215-234, 1967. · Zbl 0145.40402
[6]	S. Daneri, ”Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations,” Comm. Math. Phys., vol. 329, iss. 2, p. 745, 2014. · Zbl 1298.35140
[7]	C. De Lellis and L. Székelyhidi Jr., ”The Euler equations as a differential inclusion,” Ann. of Math., vol. 170, iss. 3, pp. 1417-1436, 2009. · Zbl 1350.35146
[8]	C. De Lellis and L. Székelyhidi Jr., ”On admissibility criteria for weak solutions of the Euler equations,” Arch. Ration. Mech. Anal., vol. 195, iss. 1, pp. 225-260, 2010. · Zbl 1192.35138
[9]	C. De Lellis and L. Székelyhidi Jr., Dissipative Euler flows and Onsager’s conjecture. · Zbl 1307.35205
[10]	C. De Lellis and L. Székelyhidi Jr., ”The \(h\)-principle and the equations of fluid dynamics,” Bull. Amer. Math. Soc., vol. 49, iss. 3, pp. 347-375, 2012. · Zbl 1254.35180
[11]	C. De Lellis and L. Székelyhidi Jr., ”Dissipative continuous Euler flows,” Invent. Math., vol. 193, iss. 2, pp. 377-407, 2013. · Zbl 1280.35103
[12]	J. Duchon and R. Robert, ”Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,” Nonlinearity, vol. 13, iss. 1, pp. 249-255, 2000. · Zbl 1009.35062
[13]	G. L. Eyink, ”Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer,” Phys. D, vol. 78, iss. 3-4, pp. 222-240, 1994. · Zbl 0817.76011
[14]	G. L. Eyink and K. R. Sreenivasan, ”Onsager and the theory of hydrodynamic turbulence,” Rev. Modern Phys., vol. 78, iss. 1, pp. 87-135, 2006. · Zbl 1205.01032
[15]	U. Frisch, Turbulence, Cambridge: Cambridge Univ. Press, 1995. · Zbl 0832.76001
[16]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001. · Zbl 1042.35002
[17]	M. Gromov, Partial Differential Relations, New York: Springer-Verlag, 1986, vol. 9. · Zbl 0651.53001
[18]	P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012. · Zbl 1367.35001
[19]	A. N. Kolmogorov, ”The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Proc. Roy. Soc. London Ser. A, vol. 434, iss. 1890, pp. 9-13, 1991. · Zbl 1142.76389
[20]	J. Nash, ”\(C^1\) isometric imbeddings,” Ann. of Math., vol. 60, pp. 383-396, 1954. · Zbl 0058.37703
[21]	L. Onsager, ”Statistical hydrodynamics,” Nuovo Cimento, vol. 6, iss. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279-287, 1949.
[22]	R. Robert, ”Statistical hydrodynamics (Onsager revisited),” in Handbook of Mathematical Fluid Dynamics, Vol. II, Amsterdam: North-Holland, 2003, pp. 1-54. · Zbl 1141.76330
[23]	V. Scheffer, ”An inviscid flow with compact support in space-time,” J. Geom. Anal., vol. 3, iss. 4, pp. 343-401, 1993. · Zbl 0836.76017
[24]	A. Shnirelman, ”Weak solutions with decreasing energy of incompressible Euler equations,” Comm. Math. Phys., vol. 210, iss. 3, pp. 541-603, 2000. · Zbl 1011.35107