Partial differential equations Equations of mathematical physics Theorem

Existence of Leray weak solutions

Builds on Navier–Stokes equations Weak solution Sobolev space

Statement

Let $\nu > 0$ and let $u_0 \in L^2(\R^3; \R^3)$ satisfy $\nabla \cdot u_0 = 0$ in the sense of distributions. Then there exists a vector field $u \in L^\infty\big(0, \infty; L^2(\R^3; \R^3)\big)$ with $\nabla u \in L^2\big(\R^3 \times (0, \infty)\big)$ such that:

$u$ is a weak solution of the Navier–Stokes equations with no external force and initial value $u_0$ . That is, $\nabla \cdot u(\cdot, t) = 0$ for almost every $t$ . Also, let $\varphi \colon \R^3 \times [0, \infty) \to \R^3$ be any smooth compactly supported test field with $\nabla \cdot \varphi = 0$ . Then $\int_0^\infty \!\! \int_{\R^3} \Big( u \cdot \partial_t \varphi + \sum_{i,j=1}^3 u_i u_j \, \partial_j \varphi_i - \nu \nabla u : \nabla \varphi \Big) dx \, dt + \int_{\R^3} u_0 \cdot \varphi(x, 0) \, dx = 0.$
$u$ satisfies the energy inequality: for every $t > 0$ , $\norm{u(\cdot, t)}_{L^2}^2 + 2 \nu \int_0^t \norm{\nabla u(\cdot, s)}_{L^2}^2 \, ds \le \norm{u_0}_{L^2}^2.$

Here $\nabla u : \nabla \varphi = \sum_{i,j} \partial_j u_i \, \partial_j \varphi_i$ . A solution with these properties is called a Leray weak solution, or a Leray–Hopf weak solution.

Remarks

Jean Leray proved this theorem in 1934, and Eberhard Hopf later proved the analogue on bounded domains. The proof regularises the nonlinear term, solves the regularised problem, and uses the energy identity to bound the approximate solutions uniformly in the Sobolev space norm. A compactness argument then gives a subsequence whose limit solves the equations in the weak sense, and the energy inequality survives the limit by lower semicontinuity of the norm. The pressure does not appear, because $\int \nabla p \cdot \varphi \, dx = 0$ for divergence-free $\varphi$ . In two space dimensions the Leray weak solution is unique and smooth for positive times. In three dimensions its uniqueness and smoothness are open, and they are the heart of the Navier–Stokes existence and smoothness problem.

Going deeper

For a student

The theorem is the starting point of the mathematical theory of the Navier–Stokes equations. The energy inequality gives the only a priori bound that is known for large data, and the class of solutions it defines is the natural one for studying regularity. A weak solution that is smooth agrees with the classical solution, and a classical solution is unique among Leray weak solutions while it exists. Results such as the partial regularity theorem of Caffarelli, Kohn and Nirenberg concern this class of solutions. Buckmaster and Vicol showed that weak solutions without the energy inequality and with less regularity are not unique.

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Navier–Stokes equations concept
Sobolev space concept
Weak solution concept
Existence of Leray weak solutions statement

Tag	`DHHR5PZZ7E`
URI	`m2://DHHR5PZZ7E`
Kind	statement, theorem
Writing	W0 stub
MSC	35Q30, 76D05, 35D30
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About	Navier–Stokes equations, Weak solution
Uses	Navier–Stokes equations, Weak solution, Sobolev space