Nonhomogeneous stochastic Navier-Stokes equations and applications.
(Equations de Navier-Stokes stochastiques non homogènes et applications.)

*(French)*Zbl 0753.35066
Tesi di Perfezionamento. Pisa: Scuola Normale Superiore, 169 p. (1992).

In this book, the author studies some problems on the existence of solutions of nonhomogeneous stochastic Navier-Stokes equations. The book is divided into five chapters.

Let be given the system \[ \rho\partial_ t u+\rho(u\cdot\nabla)u- \nu\Delta u+\nabla p=\rho f, \qquad \nabla\cdot u=0, \qquad \partial_ t\rho+\nabla\cdot(\rho u)=0,\tag{1} \] where \((u\cdot\nabla)=\sum_{i=1}^ 3 u_ i\cdot\partial_{x_ i}\), \(\nabla\cdot u=\sum_{i=1}^ 3 \partial_{x_ i}\cdot u_ i\), \(\;\nu\), \(p\), \(f\) are given functions, \(u\), \(\rho\) are unknown. The system (1) is called nonhomogeneous Navier-Stokes equations.

Consider the function spaces \(V^ \infty=\{u\in(C_ 0^ \infty({\mathcal O}))^ 3:\;\nabla\cdot u=0\) in \({\mathcal O}\}\), \(V^ 0=\text{ closure of }V^ \infty\) in \((L^ 2({\mathcal O}))^ 3\), \(V^ 1=\text{ closure of }V^ \infty\) in \((H^ 1({\mathcal O}))^ 3\), where \({\mathcal O}\subset\mathbb{R}^ 3\) is bounded and \(\partial{\mathcal O}\) is suitable regular.

In the introductory (Chapter 1), the author starts with the general definition of system (1) and recalls some basic theorems for nonhomogeneous Navier-Stokes equations. In Chapter 2, on the base of the Faedo-Galerkin method the author proves theorems on existence and uniqueness of solutions for nonhomogeneous stochastic Navier-Stokes equations.

Chapter 3, one of the main parts of the book, is devoted to the existence of weak solutions for nonhomogeneous stochastic Navier-Stokes equations. Perturbation of the right hand side in the first equation of system (1) is given by \(\rho f dt+\rho dG\), where \(G\) is a continuous stochastic process which values in \(V^ 1\). The author proves theorems utilizing some ideas contained in [A. Bensoussan and R. Temann, J. Funct. Anal. 13, 195-222 (1973; Zbl 0265.60094)].

In Chapter 4, the second main part of the book, the author investigates the problem of existence of weak solutions for nonhomogeneous stochastic Navier-Stokes equations by utilizing stochastic integration. This method allows to generalize the form of perturbation. Perturbation exit may be stochastic processes with values in \(V^ 0\).

In the last chapter (Chapter 5) the author considers the stochastic evolution equation of the form \[ du(t)+(A(t)u(t)+B(u(t)))dt=dG(t)+dH(t), \qquad u(0)=u_ 0,\tag{2} \] where \(u\) belongs to a separable Hilbert space, \(A(t)\) is a linear operator, \(B(\cdot)\) is a nonlinear operator, \(G\) is a stochastic process with bounded variation, \(H\) is a martingale with values in separable Hilbert spaces. System (2) is a generalized nonhomogeneous stochastic Navier-Stokes equation on the domain \({\mathcal O}\subset\mathbb{R}^ m\) whre \(\partial{\mathcal O}\) is suitable regular. The author proves theorems on the existence of weak solutions for system (2).

Summing up, it is a very nice and interesting book. It will surely by the research worker’s bible for many years.

Let be given the system \[ \rho\partial_ t u+\rho(u\cdot\nabla)u- \nu\Delta u+\nabla p=\rho f, \qquad \nabla\cdot u=0, \qquad \partial_ t\rho+\nabla\cdot(\rho u)=0,\tag{1} \] where \((u\cdot\nabla)=\sum_{i=1}^ 3 u_ i\cdot\partial_{x_ i}\), \(\nabla\cdot u=\sum_{i=1}^ 3 \partial_{x_ i}\cdot u_ i\), \(\;\nu\), \(p\), \(f\) are given functions, \(u\), \(\rho\) are unknown. The system (1) is called nonhomogeneous Navier-Stokes equations.

Consider the function spaces \(V^ \infty=\{u\in(C_ 0^ \infty({\mathcal O}))^ 3:\;\nabla\cdot u=0\) in \({\mathcal O}\}\), \(V^ 0=\text{ closure of }V^ \infty\) in \((L^ 2({\mathcal O}))^ 3\), \(V^ 1=\text{ closure of }V^ \infty\) in \((H^ 1({\mathcal O}))^ 3\), where \({\mathcal O}\subset\mathbb{R}^ 3\) is bounded and \(\partial{\mathcal O}\) is suitable regular.

In the introductory (Chapter 1), the author starts with the general definition of system (1) and recalls some basic theorems for nonhomogeneous Navier-Stokes equations. In Chapter 2, on the base of the Faedo-Galerkin method the author proves theorems on existence and uniqueness of solutions for nonhomogeneous stochastic Navier-Stokes equations.

Chapter 3, one of the main parts of the book, is devoted to the existence of weak solutions for nonhomogeneous stochastic Navier-Stokes equations. Perturbation of the right hand side in the first equation of system (1) is given by \(\rho f dt+\rho dG\), where \(G\) is a continuous stochastic process which values in \(V^ 1\). The author proves theorems utilizing some ideas contained in [A. Bensoussan and R. Temann, J. Funct. Anal. 13, 195-222 (1973; Zbl 0265.60094)].

In Chapter 4, the second main part of the book, the author investigates the problem of existence of weak solutions for nonhomogeneous stochastic Navier-Stokes equations by utilizing stochastic integration. This method allows to generalize the form of perturbation. Perturbation exit may be stochastic processes with values in \(V^ 0\).

In the last chapter (Chapter 5) the author considers the stochastic evolution equation of the form \[ du(t)+(A(t)u(t)+B(u(t)))dt=dG(t)+dH(t), \qquad u(0)=u_ 0,\tag{2} \] where \(u\) belongs to a separable Hilbert space, \(A(t)\) is a linear operator, \(B(\cdot)\) is a nonlinear operator, \(G\) is a stochastic process with bounded variation, \(H\) is a martingale with values in separable Hilbert spaces. System (2) is a generalized nonhomogeneous stochastic Navier-Stokes equation on the domain \({\mathcal O}\subset\mathbb{R}^ m\) whre \(\partial{\mathcal O}\) is suitable regular. The author proves theorems on the existence of weak solutions for system (2).

Summing up, it is a very nice and interesting book. It will surely by the research worker’s bible for many years.

Reviewer: S.Wedrychowicz (Rzeszów)

##### MSC:

35Q30 | Navier-Stokes equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

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

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

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35D05 | Existence of generalized solutions of PDE (MSC2000) |