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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.

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)