Existence and Smoothness of the Navier-Stokes Equation

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Scheffer [7] applied ideas from geometric measure theory to prove a partial regularity theorem for suitable weak solutions of the Navier-Stokes equations. Caffarelli-Kohn-Nirenberg [2] improved Scheffer's results, and F.-H. Lin [6] simplified the proofs of the results in Caffarelli-Kohn-Nirenberg [2]. The partial regularity theorem of [2], [6] concerns a parabolic analogue of the Hausdorff dimension of the singular set of a suitable weak solution of Navier-Stokes. Here, the singular set of a weak solution u consists of all points (x_o,t_o) in R^3 x R such that u is unbounded in every neighborhood of (x_o,t_o).

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The main results of [2], [6] may be stated roughly as follows. Theorem. (A) Let u be a weak solution of the Navier-Stokes equations, satisfying suitable growth conditions. Let E be the singular set of u. Then P_1(E) = 0. (B) Given a divergence-free vector field u_o(x) and a force f(x,t) satisfying (4) and (5), there exists a weak solution of Navier-Stokes (1), (2), (3) satisfying the growth conditions in (A). In particular, the singular set of u cannot contain a spacetime curve of the form {(x,t) in R^3 x R: x = phi(t)}. This is the best partial regularity theorem known so far for the Navier-Stokes equation.

Referencing norms: names-caffarelli-kohn-nirenberg-1982

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Equation (1) is just Newton's law f=ma for a fluid element subject to the external force f=(f_i(x,t))_{1<=i<=n} and to the forces arising from pressure and friction. Equation (2) just says that the fluid is incompressible. For physically reasonable solutions, we want to make sure u(x,t) does not grow large as |x| -> infinity. Hence, we will restrict attention to forces f and initial conditions u_o that satisfy |d^alpha_x u_o(x)| <= C_{alpha K}(1+|x|)^{-K} on R^n, for any alpha and K, (4); and |d^alpha_x d^m_t f(x,t)| <= C_{alpha m K}(1+|x|+t)^{-K} on R^n x [0,infinity), for any alpha, m, K, (5).

Referencing norms: defines-decay-conditions

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The Navier-Stokes equations are then given by (d/dt)u_i + sum_{j=1}^n u_j (d u_i/d x_j) = nu*Delta(u_i) - (d p/d x_i) + f_i(x,t) (x in R^n, t >= 0), (1); div u = sum_{i=1}^n (d u_i/d x_i) = 0 (x in R^n, t >= 0), (2); with initial conditions u(x,0) = u_o(x) (x in R^n), (3).

Referencing norms: defines-equations

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Starting with Leray [5], important progress has been made in understanding weak solutions of the Navier-Stokes equations. Leray in [5] showed that the Navier-Stokes equations (1), (2), (3) in three space dimensions always have a weak solution (p,u) with suitable growth properties. Uniqueness of weak solutions of the Navier-Stokes equation is not known.

Referencing norms: names-leray-1934

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In two dimensions, the analogues of assertions (A) and (B) have been known for a long time (Ladyzhenskaya [4]), also for the more difficult case of the Euler equations. This gives no hint about the three-dimensional case, since the main difficulties are absent in two dimensions. In three dimensions, it is known that (A) and (B) hold provided the initial velocity u_o satisfies a smallness condition. For initial data u_o(x) not assumed to be small, it is known that (A) and (B) hold (also for nu=0) if the time interval [0,infinity) is replaced by a small time interval [0,T), with T depending on the initial data. For a given initial u_o(x), the maximum allowable T is called the "blowup time." Either (A) and (B) hold, or else there is a smooth, divergence-free u_o(x) for which (1), (2), (3) have a solution with a finite blowup time. For the Navier-Stokes equations (nu>0), if there is a solution with a finite blowup time T, then the velocity (u_i(x,t))_{1<=i<=3} becomes unbounded near the blowup time.

Referencing norms: names-ladyzhenskaya-2d

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In place of (4) and (5), we assume that u_o is smooth and that |d^alpha_x d^m_t f(x,t)| <= C_{alpha m K}(1+|t|)^{-K} on R^3 x [0,infinity), for any alpha, m, K, (9).

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Alternatively, to rule out problems at infinity, we may look for spatially periodic solutions of (1), (2), (3). Thus, we assume that u_o(x), f(x,t) satisfy u_o(x+e_j) = u_o(x), f(x+e_j,t) = f(x,t) for 1 <= j <= n (e_j = j-th unit vector in R^n), (8).

Referencing norms: defines-periodic-setup

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We accept a solution of (1), (2), (3) as physically reasonable only if it satisfies p, u in C-infinity(R^n x [0,infinity)), (6); and integral over R^n of |u(x,t)|^2 dx < C for all t >= 0 (bounded energy), (7).

Referencing norms: defines-physically-reasonable-r3

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We then accept a solution of (1), (2), (3) as physically reasonable if it satisfies u(x,t) = u(x+e_j,t) on R^3 x [0,infinity) for 1 <= j <= n, (10); and p, u in C-infinity(R^n x [0,infinity)), (11).

Referencing norms: defines-physically-reasonable-torus

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A fundamental problem in analysis is to decide whether such smooth, physically reasonable solutions exist for the Navier-Stokes equations. To give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of one of the following four statements.

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The Euler and Navier-Stokes equations describe the motion of a fluid in R^n (n = 2 or 3). These equations are to be solved for an unknown velocity vector u(x,t) = (u_i(x,t))_{1<=i<=n} in R^n and pressure p(x,t) in R, defined for position x in R^n and time t >= 0. We restrict attention here to incompressible fluids filling all of R^n.

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(A) Existence and smoothness of Navier-Stokes solutions on R^3. Take nu > 0 and n = 3. Let u_o(x) be any smooth, divergence-free vector field satisfying (4). Take f(x,t) to be identically zero. Then there exist smooth functions p(x,t), u_i(x,t) on R^3 x [0,infinity) that satisfy (1), (2), (3), (6), (7).

Referencing norms: NsStatementA, a-identifies-ns, source-asserts-a

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(B) Existence and smoothness of Navier-Stokes solutions in R^3/Z^3. Take nu > 0 and n = 3. Let u_o(x) be any smooth, divergence-free vector field satisfying (8); we take f(x,t) to be identically zero. Then there exist smooth functions p(x,t), u_i(x,t) on R^3 x [0,infinity) that satisfy (1), (2), (3), (10), (11).

Referencing norms: NsStatementB, b-identifies-ns, source-asserts-b

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(C) Breakdown of Navier-Stokes solutions on R^3. Take nu > 0 and n = 3. Then there exist a smooth, divergence-free vector field u_o(x) on R^3 and a smooth f(x,t) on R^3 x [0,infinity), satisfying (4), (5), for which there exist no solutions (p,u) of (1), (2), (3), (6), (7) on R^3 x [0,infinity).

Referencing norms: NsStatementC, c-identifies-ns, source-asserts-c

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(D) Breakdown of Navier-Stokes Solutions on R^3/Z^3. Take nu > 0 and n = 3. Then there exist a smooth, divergence-free vector field u_o(x) on R^3 and a smooth f(x,t) on R^3 x [0,infinity), satisfying (8), (9), for which there exist no solutions (p,u) of (1), (2), (3), (10), (11) on R^3 x [0,infinity).

Referencing norms: NsStatementD, d-identifies-ns, source-asserts-d

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Here, u_o(x) is a given, C-infinity divergence-free vector field on R^n, f_i(x,t) are the components of a given, externally applied force (e.g. gravity), nu is a positive coefficient (the viscosity), and Delta = sum_{i=1}^n (d^2/d x_i^2) is the Laplacian in the space variables. The Euler equations are equations (1), (2), (3) with nu set equal to zero.

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EXISTENCE AND SMOOTHNESS OF THE NAVIER-STOKES EQUATION

Referencing norms: CmiNavierStokesSource, a-classified, classified_as_millennium_problem, defines_notion, names_historical_result