Borel Summability in PDE initial value problems

Transcription

1 Borel Summability in PDE initial value problems Saleh Tanveer (Ohio State University) Collaborator Ovidiu Costin & Guo Luo Research supported in part by Institute for Math Sciences (IC), EPSRC & NSF.

2 Main idea For an autonomous differential operator N, consider v t = N[v], v(x, ) = v (x) Formal small time expansion: ṽ(x, t) = v (x) + tv 1 (x) + t 2 v 2 (x) +.., where v 1 (x) = N[v ](x), v 2 = 1 2 {N v(v )[v 1 ]} (x),.. Generically divergent if order of N is greater than 1 If Borel summable, obtain v(x, t) = B ṽ, solution to the PDE initial value problem for small enough time. Depending on properties in the Borel plane, solution can be extended over longer time periods [, T].

3 Eg: 1-D Heat Equation (Lutz, Miyake & Schaefke) v t = v xx, v(x, ) = v (x), v(x, t) = v + tv Obtain recurrence relation (k + 1)v k+1 = v k, implies v k = v(2k) k! Unless v entire, series k tk v k factorially divergent. Borel transform in τ = 1/t: V (x, p) = B[v(x, 1/τ))](p), V (x, p) = p 1/2 W(x, 2 p), then W qq W xx = Obtain v(x, t) = R v (y)(4πt) 1/2 exp[ (x y) 2 /(4t)]dy, i.e. Borel sum of formal series leads to usual heat solution. We seek applications of these simple ideas to more complicated PDEs, including 3-D Navier-Stokes

4 Background Borel Summability for linear PDEs studied before (Balser, Miyake, Lutz, Schaefke,..) Sectorial existence for a class of nonlinear PDEs (Costin & T.) Complex singularity formation for a nonlinear PDE (Costin & T.) Navier Stokes is a nonlinear PDE governing fluid velocity v(x, t): v t + v v = P + ν v + f v =, v(x,) = v (x) Using PDE techniques, Leray (193s) proved local existence, uniqueness for classical solutions and global existence for weak solutions. Global existence of classical solutions known in 2-D, not in 3-D. Literature extensive ( (Constantin, Temam, Foias,...).

5 Background II Global existence of classical solution or lack of it has fundamental implications to fluid turbulence. Blow up of classical solution with finite energy v L2 (R 3 ) implies v(., t) and v(., t) L 3 (R3) blow up (Beale et al, Sverak). This becomes incompatible with the modeling assumptions in deriving Navier-Stokes. Hence other parameters not included in Navier-Stokes would become important in turbulent flow. For the usual PDE techniques, key to global existence question is believed to be a priori energy bounds involving v (Tao). None is available thus far. This motivates alternate formulation of initial value problems for nonlinear PDEs that are not dependent on energy bounds at all. Borel methods and generalization via Ecalle sum allows this.

6 Illustration: Borel Transform for Burger s equation Substitute v = v (x) + u(x, t) into v t + vv x = v xx to obtain u t u xx = v u x uv,x uu x + v 1 (x) where v 1 (x) = v v v,x, and, u(x, ) = Inverse Laplace Transform in 1/t and Fourier-Transform in x: pû pp + 2U p + k 2 Û = ˆv 1 ikˆv ˆ Û ikû Û Ĝ(k, p) + ˆv 1, ˆ is Fourier convolution, Fourier-Laplace convolution. Hence Û(k, p) = p K(p, p ; k)ĝ(k, p )dp + Û () (k, p) N [Û] (k, p) K(p, p ; k) = ikπ z {z Y 1 (z )J 1 (z) z Y 1 (z)j 1 (z )} z = 2 k p, z = 2 k p, Û () (k, p) = 2 J 1(z) ˆv 1 (k) z

7 Solution to integral equation Û = N[Û] We find K(p, p ; k) C p, C a constant ˆF(., p)ˆ Ĝ(., p) L 1 (R) C ˆF(., p) L 1 (R) Ĝ(., p) L 1 (R) Define norm. (α) for functions F(p, k) F (α) = e αp F(., p) L 1 (R) dp easily follows F G (α) C F (α) G (α) N seen to be contractive for large α implies Burgers solution for for Re 1 t > α in the form v(x, t) = v (x) + e p/t U(x, p)dp Global classical PDE solution implied if Û(., p) L1 (R 3 ) bounded. Borel summability for analytic v requires analyticity of U(., p) for p R + ; proof a bit more delicate.

8 Incompressible 3-D Navier-Stokes in Fourier-Space Consider 3-D N-S in infinite geometry or periodic box. Similar results expected for finite domain with no-slip BC using eigenfunctions of Stokes operator as basis. In Fourier-Space ˆv t + ν k 2ˆv = ik j P k [ˆv jˆ ˆv] + ˆf(k) ( P k = I k(k ) ), ˆv(k, ) = ˆv k 2 (k) where P k is the Hodge projection in Fourier space, ˆf(k) is the Fourier-Transform of forcing f(x), assumed divergence free and t-independent. Subscript j denotes the j-th component of a vector. k R 3 or Z 3. Einstein convention for repeated index followed. ˆ denotes Fourier convolution.

9 Integral equation for Navier Stokes in Borel plane Substitute ˆv = ˆv + û(k, t), into Navier-Stokes, inverse-laplace Transform in 1/t and inverting as for Burger s equation obtain integral equation: U(k, p) = p K(p, p )ˆR(k, p )dp + U () (k, p), ] ˆR(k, p) = ik j P k [ˆv,jˆ Û + Û jˆ ˆv + Û j Û U () (k, p) = 2 J 1(z) z ˆv 1 (k), where ˆv 1 (k) = k 2ˆv ik j P k [ˆv,jˆ ˆv ] + ˆf(k)

10 Some Results for Navier-Stokes (NS) in R 3 Define. µ,β, for µ > 3, β : v µ,β = sup k R 3 e β k (1 + k ) µ ˆv (k) Theorem 1: If f µ,β, ˆv µ+2,β <, NS has unique solution with ˆv(, t) µ,β < for Re 1 t > α, where α depends on ˆv, ˆf. Furthermore, ˆv(, t) is analytic for Re 1 t > α and ˆv(, t) µ+2,β < for t [, α 1 ). For β >, v is analytic in x with same analyticity width as v and f. Theorem 2: For β >, the NS solution v is Borel summable in 1/t, i.e. there exists U(x, p), analytic in a neighborhood of R +, exponentially bounded, and analytic in x for Im x < β so that v(x, t) = v (x) + U(x, p)e p/t dp. When t, v(x, t) v (x) + m=1 tm v m (x), where v m (x) m!a B m, with A, B generally dependent on v, f. Same results in T 3. Further, when v, f have finite Fourier modes, B is independent of initial data and f.

11 Define. (α) so that Results on Navier-Stokes in T 3 ˆV (α) = e αp ˆV (., p) l 1 (Z 3 )dp Theorem 3: If ˆv l 1 (Z 3 ), ˆf l 1 (Z 3 ) < then there exists some α > so [Û] that integral equation Û = N has a unique solution for p R + in the space } of functions {Û : Û (α) <. Further, ˆv(k, t) = ˆv (k) + Û(k, p)e p/t dp satisfies 3-D Navier-Stokes in Fourier-Space; corresponding v(x, t) is a classical NS solution for t (, α 1 ). Remark 1: Classical PDE methods known to give similar results. However, in the present formulation, global PDE existence is a question of asymptotics of known solution to integral equation as p. Sub-exponential growth implies global existence.

12 More Remarks on Theorem 3 for 3-D Navier-Stokes Remark 2: Errors in Numerical solutions rigorously controlled. Discretization in p and Galerkin approximation in k results in: Û δ (k, mδ) = δ m K m,m P N H δ (k, m δ) + Û () (k, mδ) m = ] N δ [Ûδ for k j = N,...N, j = 1, 2, 3 P N is the Galerkin Projection into N-Fourier modes. N δ has properties similar to N. The continuous solution Û satisfies ] Û = N δ [Û + E, where E is the truncation error. Thus, Û Û δ can be estimated using same tools as in Theorem 1. Note: Similar control over discretized solutions to PDEs not available since truncation errors involve derivatives of PDE solution which are not known to exist beyond a short-time.

13 Extending Navier-Stokes interval of existence Suppose Û(., p) is known over [, p ] through Taylor series in p or otherwise, and computed Û(., p) l 1 is observed to decrease towards the end of this interval. Prior discussions show that any error in this computation can be rigorously controlled. Results in the following page show that a more optimal Borel exponent α α may be estimated using the known solution in [, p ], where α is the initial α estimate in Theorem 1. This implies a longer interval [, α 1) for NS solution. A longer existence time for NS is relevant to the global existence question for f =, since it is known that there exists T c so that any weak Leray solution becomes classical for t > T c

14 Extending Navier-Stokes interval of existence -II For α, define ǫ = ν 1/2 p 1/2, a = ˆv l 1, c = ǫ 1 = ν 1/2 p 1/2 ( p 2 p Û () (., p) l 1e α p dp e α s Û(., s) l 1ds + ˆv l 1 ) b = e α p νp α p Û Û + ˆv Û l 1ds Theorem 4: A smooth solution to 3-D Navier-Stokes equation exists on the interval [, α 1 ), when α α is chosen to satisfy α > ǫ 1 + 2ǫc + (ǫ 1 + 2ǫc) 2 + 4bǫ ǫ 2 1

15 determines optimal α. γ gives dominant oscillation frequency. Relation of optimal α to NS time singularities Û(k, p) = 1 2πi c +i c i [ ] 1 e p/t [ˆv(k, t) ˆv (k)] d t Im 1/t α+ιγ Re 1/t α ιγ Rightmost singularity(ies) of NS solution ˆv(k, t) in the 1/t plane

16 Generalized Laplace-transform representation Since the Borel domain growth rate α relates to complex right-half 1 NS singularities, the following representation for t n > 1 is sought: ˆv(k, t) = ˆv (k) + e q/tn Û(k, q)dq Note Û(., p) Û(., q) is an Ecalle acceleration. In order that Û(., q) has no growth for large q, unless there is a NS singularity for t R +, need to know a priori that there is a singularity free sector in the right-half t-plane. This is proved to be true for f = and we have the following result: Theorem 5: For f =, if NS has a global classical solution, then for all sufficiently large n, U(x, q) = O(e C nq 1/(n+1) ) as q +, for some C n >.

17 Numerical Solutions to integral equation We choose the Kida initial conditions and forcing v (x) = (v 1 (x 1, x 2, x 3, ), v 2 (x 1, x 2, x 3, ), v 3 (x 1, x 2, x 3, )) v 1 (x 1, x 2, x 3, ) = v 2 (x 3, x 1, x 2, ) = v 3 (x 2, x 3, x 1, ) v 1 (x 1, x 2, x 3, ) = sin x 3 (cos3x 2 cos x 3 cos x 2 cos 3x 3 ) f 1 (x 1, x 2, x 3 ) = 1 5 v 1(x 1, x 2, x 3, ) High Degree of Symmetry makes computationally less expensive Corresponding Euler problem believed to blow up in finite time; so good candidate to study viscous effects In the plots, "constant forcing" corresponds to f = (f 1, f 2, f 3 ) as above, while zero forcing refers to f =. Recall sub-exponential growth in p corresponds to global N-S solution.

18 Numerical solution to integral equation-plot-1 Constant forcing kn p Û(., p) l 1 vs. p for ν = 1, constant forcing.

19 Numerical solution to integral equation-plot-2 Zero forcing b p Û(., p) l 1 vs. p for ν = 1, no forcing

20 Numerical solution to integral equation-plot-3 18 Constant forcing kn p Û(., p) l 1 vs. p for ν =.16, constant forcing

21 Numerical solution to integral equation-plot-4 18 Constant forcing kn p Û(., p) l 1 vs. p for ν =.1, constant forcing

22 Numerical solution to integral equation-plot x 1 3 k=(1,1,17) 1.5 fu p Û(k, p) vs. p for k = (1, 1, 17), ν =.1, no forcing.

23 Numerical solution to integral equation-plot-6 2 Constant forcing log(kn) p log Û(., p) l 1 vs. log p for ν =.1, constant forcing

24 Û(., q) l 1 vs. q, n = 2, ν =.1 9 Zero forcing kn q Kida I.C. v () 1 = sin x 1 (cos3x 2 cos x 3 cos x 2 cos3x 3 ) Other components from cyclic relation: v () 1 (x 1, x 2, x 3 ) = v () 1 (x 3, x 1, x 2 ) = v () 3 (x 2, x 3, x 1 )

25 [, q ] decays with q. Then α can be chosen rather small. Extending Navier-Stokes interval of existence For α, define ǫ 1 = ν 1/2 q 1+1/(2n), c = ǫ 1 = ν 1/2 q 1+1/(2n) ( q 2 q Û () (., q) l 1e α q dq e α s Û(., s) l 1ds + ˆv l 1 ) b = e α q νq 1 1/(2n) α q Û Û + ˆv Û l 1ds Theorem 6: A smooth solution to 3-D Navier-Stokes equation exists in the. l 1 space on the interval [, α 1/n ), when α α is chosen to satisfy α > ǫ 1 + 2ǫc + (ǫ 1 + 2ǫc) 2 + 4bǫ ǫ 2 1 Remark: If q is chosen large enough, ǫ, ǫ 1 is small when computed solution in

26 Other problems where approach is applicable Navier-Stokes with temperature field (Boussinesq approximation) Fourth order Parabolic equations of the type: u t + 2 u = N[u, Du, D 2 u, D 3 u] Magneto-hydrodynamic equation with certain approximations. For some PDE problems with finite-time blow-up, blow-up time related to exponent α of exponential growth of Integral equation as n.

27 Conclusions We have shown how Borel summation methods provides an alternate existence theory for PDE Initial value problems like N-S. With this integral equation (IE) approach, the PDE global existence is implied if known solution to IE has subexponential growth at. The solution to integral equation in a finite interval can be computed numerically with rigorously controlled errors. Integral equation in a suitable accelerated variable q will decay exponentially for unforced N-S equation, unless there is a real time singularity of PDE solution. The computation over a finite [, q ] interval gives a refined bound on exponent α at, and hence a longer existence time [, α 1/n ) to 3-D Navier-Stokes. Approach is applicable to a wide class of other PDE initial value problems.