On the Boundary Partial Regularity for the incompressible Navier-Stokes Equations

Transcription

1 On the for the incompressible Navier-Stokes Equations Jitao Liu Federal University Rio de Janeiro Joint work with Wendong Wang and Zhouping Xin Rio, Brazil, May

2 Outline Introduction 1 Introduction 2 Main results Difficulties Sketch of proof

3 Partial Regularity Introduction Recall the time-dependent incompressible Navier-Stokes equations { t u u + u u + π = f, (1) u = 0, and the steady incompressible Navier-Stokes equations on Ω R n, (SNS) { u + u u = π + f, u = 0. (2) u: velocity field, π: pressure, f : forcing term.

4 Global regularity of solutions to the time-dependent Navier-Stokes equations in three space dimensions is still a wide-open problem. V. Scheffer (1976 Pacific J.Math,1977 CMP,1980 CMP) established various partial regularity results for weak solutions to the 3D Navier-Stokes equations satisfying the so-called local energy inequality. L. A. Caffarelli, R. Kohn and L. Nirenberg (1982 CPAM) firstly introdued the notion of suitable weak solutions. They also proved that, for any suitable weak solution (u, π), there is an open subset where the velocity field u is regular, and showed that the 1D Hausdorff measure of the complement of this subset is equal to zero. Remark: The notions of weak solutions and suitable weak solutions essentially differ in that the latter notion also requires a generalized energy inequality and an estimate for the pressure π.

5 Global regularity of solutions to the time-dependent Navier-Stokes equations in three space dimensions is still a wide-open problem. V. Scheffer (1976 Pacific J.Math,1977 CMP,1980 CMP) established various partial regularity results for weak solutions to the 3D Navier-Stokes equations satisfying the so-called local energy inequality. L. A. Caffarelli, R. Kohn and L. Nirenberg (1982 CPAM) firstly introdued the notion of suitable weak solutions. They also proved that, for any suitable weak solution (u, π), there is an open subset where the velocity field u is regular, and showed that the 1D Hausdorff measure of the complement of this subset is equal to zero. Remark: The notions of weak solutions and suitable weak solutions essentially differ in that the latter notion also requires a generalized energy inequality and an estimate for the pressure π.

6 Global regularity of solutions to the time-dependent Navier-Stokes equations in three space dimensions is still a wide-open problem. V. Scheffer (1976 Pacific J.Math,1977 CMP,1980 CMP) established various partial regularity results for weak solutions to the 3D Navier-Stokes equations satisfying the so-called local energy inequality. L. A. Caffarelli, R. Kohn and L. Nirenberg (1982 CPAM) firstly introdued the notion of suitable weak solutions. They also proved that, for any suitable weak solution (u, π), there is an open subset where the velocity field u is regular, and showed that the 1D Hausdorff measure of the complement of this subset is equal to zero. Remark: The notions of weak solutions and suitable weak solutions essentially differ in that the latter notion also requires a generalized energy inequality and an estimate for the pressure π.

7 Global regularity of solutions to the time-dependent Navier-Stokes equations in three space dimensions is still a wide-open problem. V. Scheffer (1976 Pacific J.Math,1977 CMP,1980 CMP) established various partial regularity results for weak solutions to the 3D Navier-Stokes equations satisfying the so-called local energy inequality. L. A. Caffarelli, R. Kohn and L. Nirenberg (1982 CPAM) firstly introdued the notion of suitable weak solutions. They also proved that, for any suitable weak solution (u, π), there is an open subset where the velocity field u is regular, and showed that the 1D Hausdorff measure of the complement of this subset is equal to zero. Remark: The notions of weak solutions and suitable weak solutions essentially differ in that the latter notion also requires a generalized energy inequality and an estimate for the pressure π.

8 The key idea of CKN is to establish certain ɛ-regularity criteria, which have many other important applications, for examples: They were crucially used by L. Escauriaza,G. Seregin and V. Sverák (2003 Russian Math. Surveys) to prove the regularity of L 3, solutions to the 3D Navier-Stokes equations; They were recently by H. Jia and V. Sverák (2013 Invent. Math) to construct forward self-similar solutions from arbitrary (-1)-homogeneous initial data; Simplified proofs and improvements can be found in many works by Lin (1998 CPAM), Ladyzhenskaya-Seregin (2002 JMFM), Tian-Xin (1999 Comm. Anal. Geom.), Seregin (2007 JMS), Gustafson-Kang-Tsai (2007 CMP), Vasseur (2007 NODEA), Kukavica (2008 DCDS), Wang-Zhang (2013 Journal d Analyse Mathematique) and the references therein.

9 The key idea of CKN is to establish certain ɛ-regularity criteria, which have many other important applications, for examples: They were crucially used by L. Escauriaza,G. Seregin and V. Sverák (2003 Russian Math. Surveys) to prove the regularity of L 3, solutions to the 3D Navier-Stokes equations; They were recently by H. Jia and V. Sverák (2013 Invent. Math) to construct forward self-similar solutions from arbitrary (-1)-homogeneous initial data; Simplified proofs and improvements can be found in many works by Lin (1998 CPAM), Ladyzhenskaya-Seregin (2002 JMFM), Tian-Xin (1999 Comm. Anal. Geom.), Seregin (2007 JMS), Gustafson-Kang-Tsai (2007 CMP), Vasseur (2007 NODEA), Kukavica (2008 DCDS), Wang-Zhang (2013 Journal d Analyse Mathematique) and the references therein.

10 The key idea of CKN is to establish certain ɛ-regularity criteria, which have many other important applications, for examples: They were crucially used by L. Escauriaza,G. Seregin and V. Sverák (2003 Russian Math. Surveys) to prove the regularity of L 3, solutions to the 3D Navier-Stokes equations; They were recently by H. Jia and V. Sverák (2013 Invent. Math) to construct forward self-similar solutions from arbitrary (-1)-homogeneous initial data; Simplified proofs and improvements can be found in many works by Lin (1998 CPAM), Ladyzhenskaya-Seregin (2002 JMFM), Tian-Xin (1999 Comm. Anal. Geom.), Seregin (2007 JMS), Gustafson-Kang-Tsai (2007 CMP), Vasseur (2007 NODEA), Kukavica (2008 DCDS), Wang-Zhang (2013 Journal d Analyse Mathematique) and the references therein.

11 The key idea of CKN is to establish certain ɛ-regularity criteria, which have many other important applications, for examples: They were crucially used by L. Escauriaza,G. Seregin and V. Sverák (2003 Russian Math. Surveys) to prove the regularity of L 3, solutions to the 3D Navier-Stokes equations; They were recently by H. Jia and V. Sverák (2013 Invent. Math) to construct forward self-similar solutions from arbitrary (-1)-homogeneous initial data; Simplified proofs and improvements can be found in many works by Lin (1998 CPAM), Ladyzhenskaya-Seregin (2002 JMFM), Tian-Xin (1999 Comm. Anal. Geom.), Seregin (2007 JMS), Gustafson-Kang-Tsai (2007 CMP), Vasseur (2007 NODEA), Kukavica (2008 DCDS), Wang-Zhang (2013 Journal d Analyse Mathematique) and the references therein.

12 Introduction Noticed that time corresponds to two space dimensions according to dimensions table, for 5D steady INS, Michael Struwe (CPAM 1988) obtained Theorem Let Ω be an open domain in R 5 and f L q (Ω) for some q > 5 2. There exists an absolute constant ɛ 0 > 0 such that the following holds true: if u H 1 (Ω, R 5 ) is a weak solution to (2) which satisfies a generalized energy inequality; and if for some x 0 Ω, there is R 0 > 0 such that r 1 u 2 dx ɛ 0, r (0, R 0 ), x x 0 <r then u is Hölder continuous in a neighborhood of x 0.

13 It is also mentioned in Struwe s paper that: It would be interesting to know if analogous partial regularity results hold in dimensions n > 5. Hongjie Dong and Rober M. Strain (2012 Indiana. Univ. Math. J) obtained the interior partial regularity for the 6D Steady Navier-Stokes Equations. Their main idea is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic regularity theory. The proofs therefore do not involve any compactness argument. What about the boundary partial regularity? up to the boundary was studied by V. Scheffer (1982 CMP) for the 3D time-dependent Navier-Stokes equations, who proved that at each time slice, u is locally bounded up to the boundary except for a closed set whose 1D Hausdorff measure is finite.

14 It is also mentioned in Struwe s paper that: It would be interesting to know if analogous partial regularity results hold in dimensions n > 5. Hongjie Dong and Rober M. Strain (2012 Indiana. Univ. Math. J) obtained the interior partial regularity for the 6D Steady Navier-Stokes Equations. Their main idea is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic regularity theory. The proofs therefore do not involve any compactness argument. What about the boundary partial regularity? up to the boundary was studied by V. Scheffer (1982 CMP) for the 3D time-dependent Navier-Stokes equations, who proved that at each time slice, u is locally bounded up to the boundary except for a closed set whose 1D Hausdorff measure is finite.

15 It is also mentioned in Struwe s paper that: It would be interesting to know if analogous partial regularity results hold in dimensions n > 5. Hongjie Dong and Rober M. Strain (2012 Indiana. Univ. Math. J) obtained the interior partial regularity for the 6D Steady Navier-Stokes Equations. Their main idea is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic regularity theory. The proofs therefore do not involve any compactness argument. What about the boundary partial regularity? up to the boundary was studied by V. Scheffer (1982 CMP) for the 3D time-dependent Navier-Stokes equations, who proved that at each time slice, u is locally bounded up to the boundary except for a closed set whose 1D Hausdorff measure is finite.

16 It is also mentioned in Struwe s paper that: It would be interesting to know if analogous partial regularity results hold in dimensions n > 5. Hongjie Dong and Rober M. Strain (2012 Indiana. Univ. Math. J) obtained the interior partial regularity for the 6D Steady Navier-Stokes Equations. Their main idea is to first establish a weak decay estimate of certain scale-invariant quantities, and then successively improve this decay estimate by a bootstrap argument and the elliptic regularity theory. The proofs therefore do not involve any compactness argument. What about the boundary partial regularity? up to the boundary was studied by V. Scheffer (1982 CMP) for the 3D time-dependent Navier-Stokes equations, who proved that at each time slice, u is locally bounded up to the boundary except for a closed set whose 1D Hausdorff measure is finite.

17 CKN s partial regularity results for the 3D time-dependent Navier-Stokes equations was extended up to the flat boundary by G. Seregin (2002 JMFM) and to the C 2 boundary by G. Seregin, T. Shilkin, V. Solonnikov, (2006 J. Math. Sci. (N. Y.)). In the 5D stationary case, K. Kang (2006 JMFM) proved the partial regularity up to the boundary, which extended the interior result by Struwe. Natural question: Can we get the boundary partial regularity of 6D Steady-state Navier-Stokes Equations? Yes! Notice: The previous blow-up arguments by Seregin or Kang based on the compact imbedding W 1,2 (B 1 ) L 3 (B 1 ) which fails in the 6D case don t work here.

18 CKN s partial regularity results for the 3D time-dependent Navier-Stokes equations was extended up to the flat boundary by G. Seregin (2002 JMFM) and to the C 2 boundary by G. Seregin, T. Shilkin, V. Solonnikov, (2006 J. Math. Sci. (N. Y.)). In the 5D stationary case, K. Kang (2006 JMFM) proved the partial regularity up to the boundary, which extended the interior result by Struwe. Natural question: Can we get the boundary partial regularity of 6D Steady-state Navier-Stokes Equations? Yes! Notice: The previous blow-up arguments by Seregin or Kang based on the compact imbedding W 1,2 (B 1 ) L 3 (B 1 ) which fails in the 6D case don t work here.

19 CKN s partial regularity results for the 3D time-dependent Navier-Stokes equations was extended up to the flat boundary by G. Seregin (2002 JMFM) and to the C 2 boundary by G. Seregin, T. Shilkin, V. Solonnikov, (2006 J. Math. Sci. (N. Y.)). In the 5D stationary case, K. Kang (2006 JMFM) proved the partial regularity up to the boundary, which extended the interior result by Struwe. Natural question: Can we get the boundary partial regularity of 6D Steady-state Navier-Stokes Equations? Yes! Notice: The previous blow-up arguments by Seregin or Kang based on the compact imbedding W 1,2 (B 1 ) L 3 (B 1 ) which fails in the 6D case don t work here.

20 CKN s partial regularity results for the 3D time-dependent Navier-Stokes equations was extended up to the flat boundary by G. Seregin (2002 JMFM) and to the C 2 boundary by G. Seregin, T. Shilkin, V. Solonnikov, (2006 J. Math. Sci. (N. Y.)). In the 5D stationary case, K. Kang (2006 JMFM) proved the partial regularity up to the boundary, which extended the interior result by Struwe. Natural question: Can we get the boundary partial regularity of 6D Steady-state Navier-Stokes Equations? Yes! Notice: The previous blow-up arguments by Seregin or Kang based on the compact imbedding W 1,2 (B 1 ) L 3 (B 1 ) which fails in the 6D case don t work here.

21 Main results Introduction Main results Difficulties Sketch of proof Motivated by the bootstrap argument due to Dong-Strain (2012), Jitao Liu, Wendong Wang and Zhouping Xin (2013) can get an alternative proof of Seregin or Kang s results without using any compactness argument to obtain: Theorem Let (u, π) be a suitable weak solution to (2) in B 1 + near the boundary {x B 1, x 6 = 0}. Then 0 is a regular point of u, if there exists a small positive constant ε such that one of the following conditions holds, i) lim sup r 3 r 0 + ii) lim sup r 2 r 0 + B + r B + r u(x) 3 dx < ε, u(x) 2 dx < ε.

22 Main results Introduction Main results Difficulties Sketch of proof Motivated by the bootstrap argument due to Dong-Strain (2012), Jitao Liu, Wendong Wang and Zhouping Xin (2013) can get an alternative proof of Seregin or Kang s results without using any compactness argument to obtain: Theorem Let (u, π) be a suitable weak solution to (2) in B 1 + near the boundary {x B 1, x 6 = 0}. Then 0 is a regular point of u, if there exists a small positive constant ε such that one of the following conditions holds, i) lim sup r 3 r 0 + ii) lim sup r 2 r 0 + B + r B + r u(x) 3 dx < ε, u(x) 2 dx < ε.

23 Difficulties Introduction Main results Difficulties Sketch of proof Difficulties: The control of the pressure associated with u. Interior case: The pressure can be decomposed as a sum of a harmonic function and a term which can be easily controlled in terms of u by the Calderón-Zygmund estimate. Boundary case: There are slow decaying terms involving E 1/2 (ρ) in the pressure decomposition in the presence of boundaries, where E(ρ) = ρ 2 u(x) 2 dx. This means that E 1/2 (ρ) and D 1 (ρ) ρ 2 π L 6 5 (B + ρ ) are B + ρ the same order in the standard iterative scheme, which seems impossible to obtain an effective iterative estimate by the local energy inequality as in Dong-Strain (2012).

24 Difficulties Introduction Main results Difficulties Sketch of proof Difficulties: The control of the pressure associated with u. Interior case: The pressure can be decomposed as a sum of a harmonic function and a term which can be easily controlled in terms of u by the Calderón-Zygmund estimate. Boundary case: There are slow decaying terms involving E 1/2 (ρ) in the pressure decomposition in the presence of boundaries, where E(ρ) = ρ 2 u(x) 2 dx. This means that E 1/2 (ρ) and D 1 (ρ) ρ 2 π L 6 5 (B + ρ ) are B + ρ the same order in the standard iterative scheme, which seems impossible to obtain an effective iterative estimate by the local energy inequality as in Dong-Strain (2012).

25 Difficulties Introduction Main results Difficulties Sketch of proof Difficulties: The control of the pressure associated with u. Interior case: The pressure can be decomposed as a sum of a harmonic function and a term which can be easily controlled in terms of u by the Calderón-Zygmund estimate. Boundary case: There are slow decaying terms involving E 1/2 (ρ) in the pressure decomposition in the presence of boundaries, where E(ρ) = ρ 2 u(x) 2 dx. This means that E 1/2 (ρ) and D 1 (ρ) ρ 2 π L 6 5 (B + ρ ) are B + ρ the same order in the standard iterative scheme, which seems impossible to obtain an effective iterative estimate by the local energy inequality as in Dong-Strain (2012).

26 Sketch of proof Introduction Main results Difficulties Sketch of proof Recall the local energy inequality 2 u 2 [ φdx u 2 φ + u φ( u 2 + 2π) ] + 2fuφdx Ω Ω for any nonnegative C test function φ vanishing at the boundary and scale-invariant quantities: A(r) = r 4 u(x) 2 dx, C(r) = r 3 u(x) 3 dx, B r + B r + E(r) = r 2 u(x) 2 dx, D 1 (r) = r 2 π 6, L 5 (B + r ) D(r) = r 3 F (r) = r 3 B + r B + r B + r π π B + r 3 2 dx, πb + r = 1 B + r f (x) 3 dx. B + r πdx,

27 Sketch of proof Introduction Main results Difficulties Sketch of proof Recall the local energy inequality 2 u 2 [ φdx u 2 φ + u φ( u 2 + 2π) ] + 2fuφdx Ω Ω for any nonnegative C test function φ vanishing at the boundary and scale-invariant quantities: A(r) = r 4 u(x) 2 dx, C(r) = r 3 u(x) 3 dx, B r + B r + E(r) = r 2 u(x) 2 dx, D 1 (r) = r 2 π 6, L 5 (B + r ) D(r) = r 3 F (r) = r 3 B + r B + r B + r π π B + r 3 2 dx, πb + r = 1 B + r f (x) 3 dx. B + r πdx,

28 Main results Difficulties Sketch of proof To Overcome Difficulties: We first derive a revised local energy inequality. Interior case: They choose φ = Γζ with ζ be a cut-off function 1 and Γ = (r 2 + x 2 ) 2. Boundary case: We choose 1 Γ = (k 2 r 2 + x 2 ) 2, 1 k ρ r with k be a free constant to be determined. Then we can get a revised loacl energy inequality k 2 A(r) + E(r) C 0 k 4( r ) 2A(ρ) + C0 k 1( ρ) 3 [ 1 2 C(ρ) + C 3 (ρ)d 3 (ρ)] ρ r (ρ) 2C 1 1 +C 0 3 (ρ)f 3 (ρ), r where 1 k ρ r and the constant C 0 is independent of k, r, ρ.

29 Main results Difficulties Sketch of proof To Overcome Difficulties: We first derive a revised local energy inequality. Interior case: They choose φ = Γζ with ζ be a cut-off function 1 and Γ = (r 2 + x 2 ) 2. Boundary case: We choose 1 Γ = (k 2 r 2 + x 2 ) 2, 1 k ρ r with k be a free constant to be determined. Then we can get a revised loacl energy inequality k 2 A(r) + E(r) C 0 k 4( r ) 2A(ρ) + C0 k 1( ρ) 3 [ 1 2 C(ρ) + C 3 (ρ)d 3 (ρ)] ρ r (ρ) 2C 1 1 +C 0 3 (ρ)f 3 (ρ), r where 1 k ρ r and the constant C 0 is independent of k, r, ρ.

30 Main results Difficulties Sketch of proof To Overcome Difficulties: We first derive a revised local energy inequality. Interior case: They choose φ = Γζ with ζ be a cut-off function 1 and Γ = (r 2 + x 2 ) 2. Boundary case: We choose 1 Γ = (k 2 r 2 + x 2 ) 2, 1 k ρ r with k be a free constant to be determined. Then we can get a revised loacl energy inequality k 2 A(r) + E(r) C 0 k 4( r ) 2A(ρ) + C0 k 1( ρ) 3 [ 1 2 C(ρ) + C 3 (ρ)d 3 (ρ)] ρ r (ρ) 2C 1 1 +C 0 3 (ρ)f 3 (ρ), r where 1 k ρ r and the constant C 0 is independent of k, r, ρ.

31 Main results Difficulties Sketch of proof To Overcome Difficulties: We first derive a revised local energy inequality. Interior case: They choose φ = Γζ with ζ be a cut-off function 1 and Γ = (r 2 + x 2 ) 2. Boundary case: We choose 1 Γ = (k 2 r 2 + x 2 ) 2, 1 k ρ r with k be a free constant to be determined. Then we can get a revised loacl energy inequality k 2 A(r) + E(r) C 0 k 4( r ) 2A(ρ) + C0 k 1( ρ) 3 [ 1 2 C(ρ) + C 3 (ρ)d 3 (ρ)] ρ r (ρ) 2C 1 1 +C 0 3 (ρ)f 3 (ρ), r where 1 k ρ r and the constant C 0 is independent of k, r, ρ.

32 Main results Difficulties Sketch of proof Lemma Taking k large enough, we can find a new θ 1 2 A(θρ) + E(θρ) + θ D 2 1 (θρ) such that following uniform estimate holds. Suppose ρ > 0 is a constant. Then we can find a θ 0 small, where θ 0 does not depend on ρ, such that θ A(θ 0 ρ) + E(θ 0 ρ) + θ D1(θ 2 0 ρ) 1 [ 1 θ A(ρ) + E(ρ) + 1 θ D1(ρ) 2 ] + C 0 E 3 2 (ρ) +C 0 ( E 2 (ρ) + F 2 3 (ρ) ), where C 0 is a constant independent of ρ. This yields an effective iteration scheme.

33 Main results Difficulties Sketch of proof Lemma Taking k large enough, we can find a new θ 1 2 A(θρ) + E(θρ) + θ D 2 1 (θρ) such that following uniform estimate holds. Suppose ρ > 0 is a constant. Then we can find a θ 0 small, where θ 0 does not depend on ρ, such that θ A(θ 0 ρ) + E(θ 0 ρ) + θ D1(θ 2 0 ρ) 1 [ 1 θ A(ρ) + E(ρ) + 1 θ D1(ρ) 2 ] + C 0 E 3 2 (ρ) +C 0 ( E 2 (ρ) + F 2 3 (ρ) ), where C 0 is a constant independent of ρ. This yields an effective iteration scheme.

34 Main results Difficulties Sketch of proof Lemma Taking k large enough, we can find a new θ 1 2 A(θρ) + E(θρ) + θ D 2 1 (θρ) such that following uniform estimate holds. Suppose ρ > 0 is a constant. Then we can find a θ 0 small, where θ 0 does not depend on ρ, such that θ A(θ 0 ρ) + E(θ 0 ρ) + θ D1(θ 2 0 ρ) 1 [ 1 θ A(ρ) + E(ρ) + 1 θ D1(ρ) 2 ] + C 0 E 3 2 (ρ) +C 0 ( E 2 (ρ) + F 2 3 (ρ) ), where C 0 is a constant independent of ρ. This yields an effective iteration scheme.

35 Main results Difficulties Sketch of proof Thanks for your attention!