| Summary: | We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>. This solution family is equivalent to a fractal curve in complex space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>d</mi></msup></semantics></math></inline-formula> with random steps parametrized by <i>N</i> Ising variables <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mi>i</mi></msub><mo>=</mo><mo>±</mo><mn>1</mn></mrow></semantics></math></inline-formula>, in addition to a rational number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mi>p</mi><mi>q</mi></mfrac></semantics></math></inline-formula> and an integer winding number <i>r</i>, related by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∑</mo><msub><mi>σ</mi><mi>i</mi></msub><mo>=</mo><mi>q</mi><mi>r</mi></mrow></semantics></math></inline-formula>. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">R</mi><mi>d</mi></msub></semantics></math></inline-formula> space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo stretchy="false">→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> or its chemical potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The system with fixed <i>N</i> has different asymptotics at odd and even <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo stretchy="false">→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, but the limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></semantics></math></inline-formula> is well defined. The energy dissipation rate is analytically calculated as a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> using methods of number theory. It grows as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>/</mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> in the continuum limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></semantics></math></inline-formula>, leading to anomalous dissipation at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>∝</mo><msqrt><mi>ν</mi></msqrt><mo stretchy="false">→</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mo>−</mo><mi>λ</mi></mrow></msup></semantics></math></inline-formula>, with a continuous spectrum of indexes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> in the local limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The spectrum is determined by a resolvent, which is represented as an infinite product of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>⊗</mo><mn>3</mn></mrow></semantics></math></inline-formula> matrices depending of the element of the Euler ensemble.
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