Mathematical analysis of the stationary Navier-Stokes equations

Mathematical analysis of the Department of Mathematics, Sogang University, Republic of Korea The 3rd GCOE International Symposium Weaving Science Web beyond Particle Matter Hierarchy February 17-19, 2011, Sendai, Japan

Contents The Definition of q-weak solutions Existence and uniqueness results for bounded domains Existence and uniqueness results for exterior problems

The Let be a bounded or exterior domain in R 3 with smooth boundary. The motion of an incompressible homogeneous viscous Newtonian fluid in is described by the following nonlinear system of partial differential equations, named after Navier (1822) and Stokes (1845 ): { v t ν v + (v )v + p = f in (0, ) div v = 0 in (0, ). Notations: f = (f 1 (x, t), f 2 (x, t), f 3 (x, t)) : the external force ν > 0 : the viscosity constant v = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) : the (unknown) velocity p = p(x, t) : the (unknown) pressure ( 3 ) (v )v = v i v = (v v 1, v v 2, v v 3 ) i=1 xi

The Assume that f(x, t) f (x) = div F (x) as t for some matrix-valued field F. Then the flow fields v and p will be stabilized for large time t, i.e., v(x, t) v (x), p(x, t) p (x) as t. The limiting fields v and p satisfy the : { ν v + (v )v + p = div F in div v = 0 in. The limiting velocity v should satisfy the (no-slip or Dirichlet) boundary condition: v (x) = 0 for all x If is an exterior domain, we also need to impose the velocity at infinity: v (x) c as x where c is a constant vector.

The We consider only the Dirichlet problem for the with ν = 1: v + (v )v + p = div F in div v = 0 in (NS) v = 0 on if is bounded, and v + (v )v + p = div F in div v = 0 in v = 0 on v(x) c as x if is exterior. (NS) Here F : R 3 3, c R 3 are given; v : R 3, p : R are unknowns.

Definition of q-weak solutions For simplicity, let be bounded. Standard function spaces Let 1 < q <. Lebesgue spaces: [ 1/q f q = f q; = f(x) dx] q, L q () = { f : R f q; < }, L q () = [L q ()] 3 or [L q ()] 3 3. Remark. If 1 < q 1 < q 2 <, then L q 2 () L q 1 () L q 2 () L q 1 ()() and v q1 C() v q2.

Definition of q-weak solutions Sobolev spaces: W 1,q () = {v L q () v L q ()}, W 1,q 0 () = { v W 1,q () v = 0 on }, { } W 1,q 0,σ () = v W 1,q 0 () div v = 0 in. Remark. W 1,q 0,σ () is a Banach space (complete normed linear space) equipped with the norm [ 1/q v 1,q = v 1,q; = ( v(x) q + v(x) q ) dx]. In particular, W 1,2 0,σ () is a Hilbert space. Spaces of test functions: C 0 () = {Φ C () Φ = 0 near }.

Definition of q-weak solutions q-weak solutions Let (v, p) be a smooth solution of (NS). Then for all Φ C 0 (), ( v + (v )v + p) Φ dx = div F Φ dx and so ( v : Φ + (v )v Φ) dx p div Φ dx = F : Φ dx, which justifies Definition. A pair (v, p) is called a q-weak solution of (NS) if and v W 1,q 0,σ (), p Lq () ( v : Φ + (v )v Φ) dx p div Φ dx = F : Φ dx. for all Φ C 0 (). A 2-weak solution of (NS) will be called simply a weak solution.

Existence and uniqueness results for bounded domains Let be a bounded domain in R 3 with smooth boundary. The fundamental L 2 -result of J. Leray Theorem. [Leray, 1933] (i) (Existence) For each F L 2 (), there exists at least one weak solution of (NS). (ii) (Uniqueness) There is a small number δ > 0 such that if F satisfies F 2 δ, then there exists at most one weak solution of (NS). (iii) (Regularity) If F is smooth, then so is any weak solution of (NS). Remark. A smallness condition on F is indeed necessary to guarantee the uniqueness of weak solutions of (NS). Remark. A similar result was also established by Leray for exterior domains.

Existence and uniqueness results for bounded domains The linear L q -result of L. Cattabriga Theorem. [Cattabriga, 1961] Let 1 < q <. Then for every F L q (), there exists a unique q-weak solution (v, p) of the Stokes problem v + p = div F in div v = 0 in (S) v = 0 on. Why an L q -result for (NS)? More regular solutions (v, p): For 2 < q <, More general data F: For 1 < q < 2, W 1,q 0,σ () Lq () W 1,2 0,σ () L2 (). L 2 () L q (). A mission of mathematics: To extend linear results to more difficult nonlinear problems.

Existence and uniqueness results for bounded domains The inequalities due to O. Hölder and S.L. Sobolev Theorem. [Hölder, 1889] Let 1 < q, r <. Then fg L 1 () for all f L q (), g L r () if and only if Theorem. [Sobolev, 1938] 1 q + 1 r 1. Let 1 < q, r <. Then W 1,q 0 () Lr () and v r C v q for all v W 1,q 0 () if and only if 1 r + 1 3 1 q. Example. W 1,2 0 () Lr () 1 r + 1 3 1 2 r 6.

Existence and uniqueness results for bounded domains The restriction on q The definition of q-weak solutions of (NS) makes sense. (v )v Φ dx < for all v W 1,q 0 (), Φ C 0 () (v )v L 1 loc () for all v W1,q 0 () W 1,q 0 () Lr () and 1 r + 1 3 1 q and 1 q 1 3 + 1 1 q 1 q + 1 r 1 1 q + 1 r 1 3 2 q

Existence and uniqueness results for bounded domains The complete L q -result for smooth domains Theorem. Let 3/2 q <. (i) (Existence) For each F L q (), there exists at least one q-weak solution of (NS). (ii) (Uniqueness) There is a small number δ > 0 such that if F satisfies F 3/2 δ, then there exists at most one 3/2-weak solution of (NS). Remark. (i) Existence and uniqueness for 3/2 < q < : q = 2 : Leray (1933), 2 < q < : Cattabriga (1961), 3 < q < 2 : Serre (1983). 2 (ii) Existence and uniqueness for q = 3/2: { Uniqueness: Galdi-Sohr-Simader (2005), Existence: Kim (2009).

Existence and uniqueness results for bounded domains The complete L q -results for non-smooth domains Let be a bounded Lipschitz domain in R 3. Theorem. [Shen, 1995] There is a small number ε > 0 satisfying the following property: Let 3/2 ε q 3 + ε. Then for every F L q (), there exists a unique q-weak solution of (S). Theorem. [Choe-Kim, preprint] (i) (Existence) Let 3/2 q 3 + ε. Then for every F L q (), there exists at least one q-weak solution of (NS). (ii) (Uniqueness) There is a small number δ > 0 such that if F satisfies F 3/2 δ, then there exists at most one 3/2-weak solution of (NS).

Existence and uniqueness results for exterior problems A difficulty due to exterior domains Let = { x R 3 x > 1 }, and choose η C 0 (R3 ) such that Then η(x) = 1 for x < 2 and η(x) = 0 for x > 3. v(x) = 1 η(x) (x ) x should be the unique solution of the simplest exterior problem: { v = f in, v = 0 on v(x) 0 as x, where f = η. Note that f C 0 () but v W 1,q 0 () 3 2 < q <. Therefore, the weak L q -result fails to hold for the Laplace exterior problem if q = 3/2. The weak L q -result holds for the exterior Stokes problem if and only if 3/2 < q < 3, as shown by Galdi-Simader (1990) and Kozono-Sohr (1991), independently.

Existence and uniqueness results for exterior problems Weak Lebesgue spaces: For 1 < q < } L q weak {v () = : [v] q = sup t {x : v(x) > t} 1/q <. t>0 Remark. L q () L q 1 weak (), v(x) = x η(x) L3 weak () \ L3 (). An L q -result for exterior domains Let be a exterior domain in R 3 with smooth boundary. Theorem. [Kim-Kozono, preprint; Heck-Kim-Kozono, preprint] For 3/2 < q < 3, there is a small positive number δ = δ(q, ) > 0 such that if F and c satisfy F L 3/2 weak () Lq (), c R 3 and c + [F] 3/2 δ, then there exists a unique q-weak solution (v, p) of (NS) which also satisfies v c L 3 weak () and p L3/2 weak ().