##
**The Navier-Stokes equations – a neverending challenge?**
*(English)*
Zbl 0924.35100

This is a very interesting and informative research-expository and survey article on the famous Navier-Stokes equations in fluid mechanics. Sixty-five years ago, Leray (1934) first initiated a study of the problems of existence, uniqueness and of solutions of the unsteady Navier-Stokes equations. His pioneering work also dealt with conjectures, open questions and unsolved problems and provided the fundamental basis of all subsequent developments of the modern mathematical theory of the Navier-Stokes equations. During the last twenty-five years, considerable attention has been given to the mathematical theory of both unsteady and steady Navier-Stokes equations (see Teman (1977), Lions (1996, 1998), and Galdi (1998)).

The Navier-Stokes equations for an unsteady motion of an incompressible viscous fluid of constant density and kinematic viscosity \(\nu\) in a domain \(\Omega\subset\mathbb{R}^n\) and the continuity equation are \[ u_t+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f,\;\text{div} u=0 \] where \(u=u(x,t)\) is the unknown velocity field at \(x=(x_1,x_2, \dots,x_n)\) and \(p=p(x,t)\) is the unknown pressure field, and \(f\) is the external force. These equations are supplemented by an initial condition \(u(x,0)= a(x)\), \(x\in\Omega\) and the no-slip boundary condition \(U=0\) on \(\partial\Omega \times (0,T)\) provided \(\partial \Omega\neq \varphi\) and an appropriate condition of infinity if \(\Omega\) is unbounded. Note that \(n=2\) or 3 corresponds to the Navier-Stokes equations in two or three dimensions. The author first reviews on the steady Navier-Stokes system (see Galdi (1996, 1998)) \[ -\Delta u+\nabla q=f,\text{ div} u=0 \text{ on } \Omega, \;u=0 \text{ on } \partial\Omega \] and the Stokes operator \(A\) defined by \[ Au=-P\Delta u. \] A modern functional analysis treatment is used to study the Navier-Stokes system.

The unsteady Navier-Stokes system is given by \[ u_t-\Delta u+\nabla q=f, \text{ div} u=0 \text{ on }[0,T] \times \Omega,\quad u(0,x)= a(x)\text{ on }\Omega, \quad u=0 \text{ on }[0,T] \times\partial\Omega. \] It turns out that the Stokes operator \(A\) generates an analytic semigroup \(\exp(-tA)\) which is uniformly bounded. In order to study existence of a global weak solution on an arbitrary time interval \([0, T]\), Leray (1934) established the energy inequality \[ \bigl\| u(t)\bigr \|^2_2 +2\int^t_0 \bigl\| \nabla u(s) \bigr\|^2_2 ds\leq\| u_0 \|^2_2 +2\int^t_0\int_\Omega fudxds. \] This may be used as the basic estimate for a Galerkin procedure in passing to the limit of the finite-dimensional approximations.

The author next discusses the questions of uniqueness and regularity of solution of the Navier-Stokes equations. This is followed by a discussion of several theorems concerning the strong solutions, and then turbulent solutions with the generalized energy inequality (GEI) \[ \bigl\| u(t) \bigr\|^2_2 +2\int^t_s \bigl\| \nabla u(\tau)\bigr \|^2_2 d\tau \leq \bigl \| u(s)\bigr\|^2_2+2\int^t_0\langle f,u\rangle d\tau \] for almost all \(s\geq 0\) and all \(t\geq s\). Several functional analytic methods are now available to construct turbulent solutions. The GEI is not only useful for finding decay estimates, but also for local reconstruction and identification of solutions (see Heywood, 1988). Included are regularity and structure theorems. A recent important development of 1996 deals with an old question of Leray for a construction of a solution with singularity. It was shown by Leray (1934) that a self-similar solution on \(\mathbb{R}^3\times(0,T)\) of the form \(u(x,t)= \lambda(t) U(x \lambda(t))\) with \(\lambda(t)= ({2a\over T-t})^{1/2}\) exists provided \(U\) satisfies a certain partial differential system. Neças et al. (1996) proved the nonexistence of a nontrivial solution \(U\) which excludes a solution of the Navier-Stokes equations with particular type of singularity. Another observation is that \(| u|^2+2Q+2ay.U\) satisfies a maximum principle.

The final section of this article deals with the large time asymptotic behavior of solutions and stability analysis. In his 1934 seminal paper, Leray raised the questions whether the energy \(\| u\|^2_2\) of the weak solution of the Navier-Stokes equations on \(\mathbb{R}^3\) which he had constructed approaches zero as \(t\to \infty\) provided there is no external force term. Both Kato (1984) and Masuda (1984) independently provided an affirmative answer to Leray’s question and also gave some important estimates for rates of decay. Subsequently, several authors proved results for higher-order norms involving fractional powers of the Stokes operator and weighted-norm estimates. Although the fundamental problem remains unsolved, a unique global solution exists provided the initial value and the external force are small and the solution is smooth depending on the smoothness of the data. For dimension \(n=2\), the mathematical theory of the Navier-Stokes equations is completely developed (see Teman (1977)). In the opinion of the reviewer, the author’s survey article is an important contribution to the field of Navier-Stokes equations as it will help to stimulate further research on this mathematically fundamental and physically useful partial differential system.

The Navier-Stokes equations for an unsteady motion of an incompressible viscous fluid of constant density and kinematic viscosity \(\nu\) in a domain \(\Omega\subset\mathbb{R}^n\) and the continuity equation are \[ u_t+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f,\;\text{div} u=0 \] where \(u=u(x,t)\) is the unknown velocity field at \(x=(x_1,x_2, \dots,x_n)\) and \(p=p(x,t)\) is the unknown pressure field, and \(f\) is the external force. These equations are supplemented by an initial condition \(u(x,0)= a(x)\), \(x\in\Omega\) and the no-slip boundary condition \(U=0\) on \(\partial\Omega \times (0,T)\) provided \(\partial \Omega\neq \varphi\) and an appropriate condition of infinity if \(\Omega\) is unbounded. Note that \(n=2\) or 3 corresponds to the Navier-Stokes equations in two or three dimensions. The author first reviews on the steady Navier-Stokes system (see Galdi (1996, 1998)) \[ -\Delta u+\nabla q=f,\text{ div} u=0 \text{ on } \Omega, \;u=0 \text{ on } \partial\Omega \] and the Stokes operator \(A\) defined by \[ Au=-P\Delta u. \] A modern functional analysis treatment is used to study the Navier-Stokes system.

The unsteady Navier-Stokes system is given by \[ u_t-\Delta u+\nabla q=f, \text{ div} u=0 \text{ on }[0,T] \times \Omega,\quad u(0,x)= a(x)\text{ on }\Omega, \quad u=0 \text{ on }[0,T] \times\partial\Omega. \] It turns out that the Stokes operator \(A\) generates an analytic semigroup \(\exp(-tA)\) which is uniformly bounded. In order to study existence of a global weak solution on an arbitrary time interval \([0, T]\), Leray (1934) established the energy inequality \[ \bigl\| u(t)\bigr \|^2_2 +2\int^t_0 \bigl\| \nabla u(s) \bigr\|^2_2 ds\leq\| u_0 \|^2_2 +2\int^t_0\int_\Omega fudxds. \] This may be used as the basic estimate for a Galerkin procedure in passing to the limit of the finite-dimensional approximations.

The author next discusses the questions of uniqueness and regularity of solution of the Navier-Stokes equations. This is followed by a discussion of several theorems concerning the strong solutions, and then turbulent solutions with the generalized energy inequality (GEI) \[ \bigl\| u(t) \bigr\|^2_2 +2\int^t_s \bigl\| \nabla u(\tau)\bigr \|^2_2 d\tau \leq \bigl \| u(s)\bigr\|^2_2+2\int^t_0\langle f,u\rangle d\tau \] for almost all \(s\geq 0\) and all \(t\geq s\). Several functional analytic methods are now available to construct turbulent solutions. The GEI is not only useful for finding decay estimates, but also for local reconstruction and identification of solutions (see Heywood, 1988). Included are regularity and structure theorems. A recent important development of 1996 deals with an old question of Leray for a construction of a solution with singularity. It was shown by Leray (1934) that a self-similar solution on \(\mathbb{R}^3\times(0,T)\) of the form \(u(x,t)= \lambda(t) U(x \lambda(t))\) with \(\lambda(t)= ({2a\over T-t})^{1/2}\) exists provided \(U\) satisfies a certain partial differential system. Neças et al. (1996) proved the nonexistence of a nontrivial solution \(U\) which excludes a solution of the Navier-Stokes equations with particular type of singularity. Another observation is that \(| u|^2+2Q+2ay.U\) satisfies a maximum principle.

The final section of this article deals with the large time asymptotic behavior of solutions and stability analysis. In his 1934 seminal paper, Leray raised the questions whether the energy \(\| u\|^2_2\) of the weak solution of the Navier-Stokes equations on \(\mathbb{R}^3\) which he had constructed approaches zero as \(t\to \infty\) provided there is no external force term. Both Kato (1984) and Masuda (1984) independently provided an affirmative answer to Leray’s question and also gave some important estimates for rates of decay. Subsequently, several authors proved results for higher-order norms involving fractional powers of the Stokes operator and weighted-norm estimates. Although the fundamental problem remains unsolved, a unique global solution exists provided the initial value and the external force are small and the solution is smooth depending on the smoothness of the data. For dimension \(n=2\), the mathematical theory of the Navier-Stokes equations is completely developed (see Teman (1977)). In the opinion of the reviewer, the author’s survey article is an important contribution to the field of Navier-Stokes equations as it will help to stimulate further research on this mathematically fundamental and physically useful partial differential system.

Reviewer: L.Debnath (Orlando)

### MSC:

35Q30 | Navier-Stokes equations |

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

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35Q40 | PDEs in connection with quantum mechanics |