##
**Mathematical problems of statistical hydromechanics. Transl. from the Russian by D. A. Leites.**
*(English)*
Zbl 0688.35077

Mathematics and its Applications. Soviet Series 9. Dordrecht etc.: Kluwer Academic Publishers (ISBN 978-90-277-2336-9/hbk; 978-94-010-7137-6/pbk; 978-94-009-1423-0/ebook). ix, 576 p. (1988).

[See also the review of the German translation in Zbl 0626.35002).]

The Navier-Stokes equations play an important role in hydro- and aerodynamics. There are many profound results on special solutions of the Navier-Stokes system. However, at low viscosity and high velocity the flow becomes turbulent, the special solutions seem to become too complicated and a statistical description of the solution is more relevant for applications. This book deals with certain mathematical problems of statistical hydrodynamics.

In Chapter I the problem of expanding the solution of Navier-Stokes system into a converging functional-analytic series in initial values is studied. The functional spaces where the solution is decomposable into a functional-analytic series in initial values and external forces from a neighbourhood of zero are found. It is shown for Burgers equations that the radius of convergence of the above-mentioned functional-analytic expansion is finite.

In Chapter II the basic facts on measure theory are given.

Chapter III is devoted to the study of moments of spatial statistical solutions for small Reynolds numbers. In this case, unique solvability of the Cauchy problem for an infinite chain of Friedman-Keller’s moment equations is proved. The authors show that the moments of the statistical solution for \(t>0\) decompose into converging series of moments of the initial measure. For small Reynolds numbers the closure problem of infinite chain of moment equations is solved.

In Chapter IV, using Galerkin approximation of the Navier-Stokes system, a spacetime statistical solution is constructed, when the mean energy is finite at the initial moment. This solution completely describes statistically turbulent fields. If the uniqueness theorem for special solutions holds, the uniqueness theorem for statistical solutions is proved.

In Chapter V the solvability of the Cauchy problem for the Hopf equation is deduced from the solvability of the corresponding problem for the spacetime statistical solution, constructed in Chapter IV. The many-time- value statistical solution of the Navier-Stokes system is a mutual restriction at different times \(t_ 1,...,t_ k\in [0,T]\) of the spacetime statistical solution. These restrictions satisfy the chain of many-time-value Hopf equations. For this chain of equations the solvability of the Cauchy problem is proved. A spacetime statistical solution is derived from the obtained consistent family of many-time- valued statistical solutions.

In Chapter VI the moments of a statistical solution for any Reynolds numbers are studied. By the solvability theorem of the Hopf equation authors prove the solvability of an infinite chain of moment equations. For the case corresponding to the two-dimensional Navier-Stokes system the classes of uniqueness for solutions of this chain are considered. The authors suggest a method of approximation for an infinite chain connected with Galerkin approximations of the Navier-Stokes system and prove the convergence of the solutions of an approximating system to solutions of the chain of moment equations. The asymptotic expansion of moments for \(t>0\) in moments of the initial measure is also constructed.

In Chapter VII a space-homogeneous statistical solution of the Navier- Stokes system in \(\mathbb R^ n\) is constructed under the assumption that the initial measure is space-homogeneous and possesses a finite mean energy density. The mean energy is infinite in this case. The methods of probability theory used in Chapter VII lead to new theorems for individual solutions.

Chapter VIII deals with these problems. In particular the existence with probability 1 of solutions of the Cauchy problem for the Navier-Stokes system for almost-periodic initial values \(U_ 0(x)\) is proved. For an almost-periodic trigonometric polynomial \(U_ 0(x)\) the solvability of the Cauchy problem, in a.a. cases, with respect to the Lebesgue measure of the range of coefficients of the initial polynomial, is proved. The answer to A. M. Kolmogorov’s question whether space-time statistical solutions \(P^{\nu}\) of the Navier-Stokes system converge in some sense as the viscosity coefficient \(\nu\) \(\to 0\) is given. A homogeneous solution of the Cauchy problem for the Hopf equation which corresponds to the Navier-Stokes system as a restriction at \(t=\text{const}\) of the space-time statistical solution is constructed. The asymptotic behaviour, as \(| x| \to 0\), of the correlation tensor of a homogeneous statistical solution is derived.

In Chapter IX an asymptotic behaviour, as \(t\to \infty\), of the Fourier coefficients of solutions of the two-dimensional Navier-Stokes system, is derived. To construct the asymptotes, the authors make use of the functional-analytic expansions and analytic first integrals presented in Chapter IX. The asymptotic of the Fourier coefficients of moments of the spatial statistical solution for small Reynolds numbers, as \(t\to \infty\), are constructed.

In Chapter X a statistical solution of the Navier-Stokes system with white noise in a bounded domain in \(\mathbb R^ 2\) or \(\mathbb R^ 3\) is constructed. In the case \(\mathbb R^ 2\), the authors prove the uniqueness of the statistical solution and construct a measurable random process satisfying such a system.

In Chapter XI the direct and the inverse Kolmogorov equations corresponding to the Navier-Stokes system with white noise, considered in Chapter X, are studied. The authors prove the solvability of the Cauchy problem for these equations and the uniqueness of the solution for \(\mathbb R^ 2\). For the inverse equation for \(\mathbb R^ 2\) and \(\mathbb R^ 3\) the generalized solution is constructed. The classical solutions of this equation in the case \(\mathbb R^ 2\) for boundary values periodic in x are constructed too.

In Chapter XII a statistical solution, space-homogeneous with respect to x, of Navier-Stokes system with white noise in \(\mathbb R^ n\) is constructed. The results on the stabilization of homogeneous statistical solutions of nonlinear parabolic equations with white noise are expressed.

In Appendix I the results connected with the unique solvability on any time segment of a three-dimensional Navier-Stokes system and the corresponding chains of moment equations are expressed.

In Appendix II the theorems on approximations of a translationally homogeneous measure by homogeneous measures supported on trigonometric polynomials are proved. The authors also prove the possibility of a similar approximation of a measure corresponding to a Wiener process translationally homogeneous with respect to \(x\).

The Navier-Stokes equations play an important role in hydro- and aerodynamics. There are many profound results on special solutions of the Navier-Stokes system. However, at low viscosity and high velocity the flow becomes turbulent, the special solutions seem to become too complicated and a statistical description of the solution is more relevant for applications. This book deals with certain mathematical problems of statistical hydrodynamics.

In Chapter I the problem of expanding the solution of Navier-Stokes system into a converging functional-analytic series in initial values is studied. The functional spaces where the solution is decomposable into a functional-analytic series in initial values and external forces from a neighbourhood of zero are found. It is shown for Burgers equations that the radius of convergence of the above-mentioned functional-analytic expansion is finite.

In Chapter II the basic facts on measure theory are given.

Chapter III is devoted to the study of moments of spatial statistical solutions for small Reynolds numbers. In this case, unique solvability of the Cauchy problem for an infinite chain of Friedman-Keller’s moment equations is proved. The authors show that the moments of the statistical solution for \(t>0\) decompose into converging series of moments of the initial measure. For small Reynolds numbers the closure problem of infinite chain of moment equations is solved.

In Chapter IV, using Galerkin approximation of the Navier-Stokes system, a spacetime statistical solution is constructed, when the mean energy is finite at the initial moment. This solution completely describes statistically turbulent fields. If the uniqueness theorem for special solutions holds, the uniqueness theorem for statistical solutions is proved.

In Chapter V the solvability of the Cauchy problem for the Hopf equation is deduced from the solvability of the corresponding problem for the spacetime statistical solution, constructed in Chapter IV. The many-time- value statistical solution of the Navier-Stokes system is a mutual restriction at different times \(t_ 1,...,t_ k\in [0,T]\) of the spacetime statistical solution. These restrictions satisfy the chain of many-time-value Hopf equations. For this chain of equations the solvability of the Cauchy problem is proved. A spacetime statistical solution is derived from the obtained consistent family of many-time- valued statistical solutions.

In Chapter VI the moments of a statistical solution for any Reynolds numbers are studied. By the solvability theorem of the Hopf equation authors prove the solvability of an infinite chain of moment equations. For the case corresponding to the two-dimensional Navier-Stokes system the classes of uniqueness for solutions of this chain are considered. The authors suggest a method of approximation for an infinite chain connected with Galerkin approximations of the Navier-Stokes system and prove the convergence of the solutions of an approximating system to solutions of the chain of moment equations. The asymptotic expansion of moments for \(t>0\) in moments of the initial measure is also constructed.

In Chapter VII a space-homogeneous statistical solution of the Navier- Stokes system in \(\mathbb R^ n\) is constructed under the assumption that the initial measure is space-homogeneous and possesses a finite mean energy density. The mean energy is infinite in this case. The methods of probability theory used in Chapter VII lead to new theorems for individual solutions.

Chapter VIII deals with these problems. In particular the existence with probability 1 of solutions of the Cauchy problem for the Navier-Stokes system for almost-periodic initial values \(U_ 0(x)\) is proved. For an almost-periodic trigonometric polynomial \(U_ 0(x)\) the solvability of the Cauchy problem, in a.a. cases, with respect to the Lebesgue measure of the range of coefficients of the initial polynomial, is proved. The answer to A. M. Kolmogorov’s question whether space-time statistical solutions \(P^{\nu}\) of the Navier-Stokes system converge in some sense as the viscosity coefficient \(\nu\) \(\to 0\) is given. A homogeneous solution of the Cauchy problem for the Hopf equation which corresponds to the Navier-Stokes system as a restriction at \(t=\text{const}\) of the space-time statistical solution is constructed. The asymptotic behaviour, as \(| x| \to 0\), of the correlation tensor of a homogeneous statistical solution is derived.

In Chapter IX an asymptotic behaviour, as \(t\to \infty\), of the Fourier coefficients of solutions of the two-dimensional Navier-Stokes system, is derived. To construct the asymptotes, the authors make use of the functional-analytic expansions and analytic first integrals presented in Chapter IX. The asymptotic of the Fourier coefficients of moments of the spatial statistical solution for small Reynolds numbers, as \(t\to \infty\), are constructed.

In Chapter X a statistical solution of the Navier-Stokes system with white noise in a bounded domain in \(\mathbb R^ 2\) or \(\mathbb R^ 3\) is constructed. In the case \(\mathbb R^ 2\), the authors prove the uniqueness of the statistical solution and construct a measurable random process satisfying such a system.

In Chapter XI the direct and the inverse Kolmogorov equations corresponding to the Navier-Stokes system with white noise, considered in Chapter X, are studied. The authors prove the solvability of the Cauchy problem for these equations and the uniqueness of the solution for \(\mathbb R^ 2\). For the inverse equation for \(\mathbb R^ 2\) and \(\mathbb R^ 3\) the generalized solution is constructed. The classical solutions of this equation in the case \(\mathbb R^ 2\) for boundary values periodic in x are constructed too.

In Chapter XII a statistical solution, space-homogeneous with respect to x, of Navier-Stokes system with white noise in \(\mathbb R^ n\) is constructed. The results on the stabilization of homogeneous statistical solutions of nonlinear parabolic equations with white noise are expressed.

In Appendix I the results connected with the unique solvability on any time segment of a three-dimensional Navier-Stokes system and the corresponding chains of moment equations are expressed.

In Appendix II the theorems on approximations of a translationally homogeneous measure by homogeneous measures supported on trigonometric polynomials are proved. The authors also prove the possibility of a similar approximation of a measure corresponding to a Wiener process translationally homogeneous with respect to \(x\).

Reviewer: A. D. Borisenko

### MSC:

35Q30 | Navier-Stokes equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35Q99 | Partial differential equations of mathematical physics and other areas of application |