##
**An introduction to the Navier-Stokes initial-boundary value problem.**
*(English)*
Zbl 1108.35133

Galdi, Giovanni P. (ed.) et al., Fundamental directions in mathematical fluid mechanics. Basel: Birkhäuser (ISBN 3-7643-6414-9). 1-70 (2000).

From the introduction: One of the aims of this article is to furnish an elementary presentation of some of the basic results so far known for given the body force \(f\) and the initial distribution of velocity \(v_0\) (no matter have smooth), to determine a corresponding unique regular solution \(v(x,t)\), \(p(x,t)\) to (0.1)–(0.3) for all times \(t>0\).

In Section 1, we discuss the main features of system (0.1) and describe the main difficulties related to it. Successively, following the classical metbods of Leray and Hopf, we introduce the definition of weak solution to (0.1)–(0.3) and study some of the related properties (Section 2). These solutions play a major role in the mathematical theory of Navier-Stokes equations, in that they are the only solutions, so far known, which exist for all times and without restrictions on the size of the data. In Section 3 we show the existence of a weak solution for all times \(t>0\).

Uniqueness and regularity of Leray-Hopf solutions is presented in Sections 4 and 5, respectively. Due to the particular form of the nonlinearity involved in the Navier-Stokes equations, this study will naturally lead to the functional class \(L^{s,r}\equiv L^r(0,T;{\mathbf L}^s(\Omega))\), \(n/s+2/r=1\), \(s> n\), \(n\) denotes the space dimension such that any weak solution belonging to \(L^{s,r}\) is unique and regular. In view of this result, we see that every weak solution in dimension two is unique within its class, and that it possesses as much space-time regularity as allowed by the data. Since it is not known if a weak solution in dimension three is in \(L^{s,r}\), it is not known if these properties continue to hold for three-dimensional flows. However, “partial regularity” results are available.

To show some of these latter, we begin to prove the existence of more regular solutions in Sections 6. This existence theory will lead to the celebrated “théorème de structure” of Leray, which, roughly speaking, states that every weak solution is regular in space and time, with the possible exception of a set of times \(I\) of zero \(1/2\)-dimensional Hausdorff measure. Moreover, defining a finite time \(t_1\in I\) an epoch of irregularity for a weak solution \({\mathbf v}\), if \({\mathbf v}\) is regular in a left-neighborhood of \(t_1\) but it can not be extended to a regular solution after \(t_1\), we give blow-up estimates for the Dirichlet norm of \(v\) at any (possible) epoch of irregularity.

In view of the relevance of the functional class \(L^{s,r}\), in Section 7 we investigate the existence of weak solutions in such a class. Specifically we shall prove the existence of weak solutions in \(L^{s,r}\), at least for small times, provided the initial data are given in Lebesgue spaces \(L^q\), for a suitable \(q\). To avoid technical difficulties, this study will be performed for the case \(\Omega=\mathbb R^n\) (Cauchy problem). As a consequence of these results, we enlarge the class of uniqueness of weak solutions, to include the case \(s=n\). In addition, we give partial regularity results of a weak solution belonging to \(L^{n,\infty}\). The important question of whether a weak solution in \(L^{n,\infty}\) is regular, is left open.

For the entire collection see [Zbl 0948.00020].

In Section 1, we discuss the main features of system (0.1) and describe the main difficulties related to it. Successively, following the classical metbods of Leray and Hopf, we introduce the definition of weak solution to (0.1)–(0.3) and study some of the related properties (Section 2). These solutions play a major role in the mathematical theory of Navier-Stokes equations, in that they are the only solutions, so far known, which exist for all times and without restrictions on the size of the data. In Section 3 we show the existence of a weak solution for all times \(t>0\).

Uniqueness and regularity of Leray-Hopf solutions is presented in Sections 4 and 5, respectively. Due to the particular form of the nonlinearity involved in the Navier-Stokes equations, this study will naturally lead to the functional class \(L^{s,r}\equiv L^r(0,T;{\mathbf L}^s(\Omega))\), \(n/s+2/r=1\), \(s> n\), \(n\) denotes the space dimension such that any weak solution belonging to \(L^{s,r}\) is unique and regular. In view of this result, we see that every weak solution in dimension two is unique within its class, and that it possesses as much space-time regularity as allowed by the data. Since it is not known if a weak solution in dimension three is in \(L^{s,r}\), it is not known if these properties continue to hold for three-dimensional flows. However, “partial regularity” results are available.

To show some of these latter, we begin to prove the existence of more regular solutions in Sections 6. This existence theory will lead to the celebrated “théorème de structure” of Leray, which, roughly speaking, states that every weak solution is regular in space and time, with the possible exception of a set of times \(I\) of zero \(1/2\)-dimensional Hausdorff measure. Moreover, defining a finite time \(t_1\in I\) an epoch of irregularity for a weak solution \({\mathbf v}\), if \({\mathbf v}\) is regular in a left-neighborhood of \(t_1\) but it can not be extended to a regular solution after \(t_1\), we give blow-up estimates for the Dirichlet norm of \(v\) at any (possible) epoch of irregularity.

In view of the relevance of the functional class \(L^{s,r}\), in Section 7 we investigate the existence of weak solutions in such a class. Specifically we shall prove the existence of weak solutions in \(L^{s,r}\), at least for small times, provided the initial data are given in Lebesgue spaces \(L^q\), for a suitable \(q\). To avoid technical difficulties, this study will be performed for the case \(\Omega=\mathbb R^n\) (Cauchy problem). As a consequence of these results, we enlarge the class of uniqueness of weak solutions, to include the case \(s=n\). In addition, we give partial regularity results of a weak solution belonging to \(L^{n,\infty}\). The important question of whether a weak solution in \(L^{n,\infty}\) is regular, is left open.

For the entire collection see [Zbl 0948.00020].

### MSC:

35Q30 | Navier-Stokes equations |

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

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |