Robert L. Jerrardcermics.enpc.fr/~al-hajm/ile-de-re/jerrard.pdfVery weak solutions of binormal...

Very weak solutions of binormal curvature flow

Robert L. Jerrard

Department of MathematicsUniversity of Toronto

Ile de Ré,May 24, 2010

Robert L. Jerrard (Toronto ) Binormal Curvature Flow May 24, 2011 1 / 18

context: a family of problems

C-valued functions ? ?←→ codimension 2 surfaces.0 < ε 1 ε = 0

object u : Ω ⊂ RN → C Γ codim 2 in Ωor u : [0, T ]×Ω→ C or in [0, T ]×Ω

functional Eε(u) := 1| lnε|

|∇u|2

2 + 14ε2 (|u|2 − 1)2 HN−2(Γ)

elliptic eqn −∆u + 1ε2 (|u|2 − 1)u = 0 H = 0

parabolic eqn ut −∆u + 1ε2 (|u|2 − 1)u = 0 V − H = 0

hyperbolic eqn utt −∆u + 1ε2 (|u|2 − 1)u = 0 Hmink = 0

Schrödinger eqn iut −∆u + 1ε2 (|u|2 − 1)u = 0 V = ?(H ∧ τ)

aka Gross-Pitaevskii binormal mean curvatureRobert L. Jerrard (Toronto ) Binormal Curvature Flow May 24, 2011 2 / 18

what is known

C-valued functions ? ?←→ codimension 2 surfaces.

functionals J-Soner, Alberti-Baldo-Orlandi, early 00s (using J, Sandier late 90s)

elliptic regularity and asymptotics, 93-03 Riviére, Lin-Riviére, Bethuel et al...

parabolic

weighted energy ests, local in t J-Soner, Lin 96

conditional result, global in t Ambrosio-Soner 98

global in t in R3 Lin-Rivière 00

global in t , in RN Bethuel-Orlandi-Smets 05

hyperbolicconditional result, global in t : Bellettini-Novaga-Orlandi, 08

unconditional result, local in t : J 09

many open problems remain

Schrödinger still poorly understood. partial result: J 02

The connection between Gross-Pitaeskii and binormal curvature flowcan be formally seen in the following two lemmas.

Lemma (J 02)

If u : (0, T )×R3 → C solves Gross-Pitaevsky (suitably scaled in t) then

X · Ju = −

D(curl X ) :(Du ⊗ Du)

| log ε|dx

for all X ∈ C2c (R3; R3), where Ju = du1 ∧ du2, u = u1 + iu2.

Lemma (J 02)

Let γ : T× R/L Z→ R3 be a smooth binormal curvature flow. Then forany X ∈ C∞

c (R3; R3), with τ := ∂sγ denoting the unit tangent,

∫γ(t ,·)

X · τdH1 = −

∫γ(t ,·)

D (curlX ) : (τ⊗ τ) dH1,

The connection between Gross-Pitaeskii and binormal curvature flowcan be formally seen in the following two lemmas.

Lemma (J 02)

If u : (0, T )×R3 → C solves Gross-Pitaevsky (suitably scaled in t) then

X · Ju = −

D(curl X ) :(Du ⊗ Du)

| log ε|dx

for all X ∈ C2c (R3; R3), where Ju = du1 ∧ du2, u = u1 + iu2.

Lemma (J 02)

Let γ : T× R/L Z→ R3 be a smooth binormal curvature flow. Then forany X ∈ C∞

c (R3; R3), with τ := ∂sγ denoting the unit tangent,

∫γ(t ,·)

X · τdH1 = −

∫γ(t ,·)

D (curlX ) : (τ⊗ τ) dH1,

Goal of this talk

We define and study a very weak notion of solution of the binormalcurvature flow, based on Lemma 2 above.

Motivations:

see above: possibly useful for ε→ 0 limit of Gross-Pitaevsky

allows changes of topology (eg, multiple filaments that collide,merge, split apart....) (these phenomena are seen in vortices in physical

fluids...)

turns out to yield new estimates for (1-d) Schrödinger maps.

all new results joint with Didier Smets

binormal curvature flow

The equation for binormal curvature flow is:

γt = γs × γss (1)

for γ : R× (R/L Z)→ R3, t ∈ R is the time variable, L =length,s ∈ R/L Z is assumed to be an arc-length parameter, i.e.

|∂s γ(t , s)|2 = 1. (2)

Equivalently, we can write (1) as

∂t γ = κb

where κ = curvature, b =binormal. This equation

is thought to approximate vortex filament motion in certain fluids;can be thought of as a “Schrödinger equation for curves”.

known attributes I

The remarkable Hasimoto transform: Let

ψ(t , s) = κ(t , s) exp(i∫ s

T (t ,σ)dσ)

where T =torsion. Then ψ solves

iψt +ψss +12(|ψ|2 + A(t , s))ψ = 0.

We can set A(t , s) = 0 by choosing s0(t , s) suitably.Consequences of the Hasimoto transform:

well-posedness results.solitons on a vortex filament (Hasimoto 1970)infinite number of conserved quantities: arclength,

∫κ2, ...

known attributes II

Related to equation for (1-d) Schrödinger maps into S2:

ut = u × uss u : [0, T ]× R× (R/L Z)→ S2.

If γ is a binormal flow, then u = γs : R× (R/LZ)→ S2 is aSchrödinger map.

This leads to a geometric interpretation of the Hasimoto transform(Chang, Shatah, Uhlenbeck, 2000)

existence of singular, self-similar solutions of the form(t , s) 7→ γselfsim(t , s) := t

√|t |G( s√

|t|), detailed analysis by Gutiérrez, Rivas,

and Vega, 2003

stable dynamics near self-similar solutions. Related via Hasimototo NLS with initial data u0 = aδ0+perturbation. (Banica and Vega 2009)

known attributes III

MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

cos ! ! 1" 2A21: #12$

Given any angle ! we will find a solution of Eq. (10) which is C1 for all positivetime. Notice that LIA is an equation reversible in time; if X#s, t$ solves Eq. (5) sodoes "X#s,"t$. As a consequence we will obtain a one parameter family of smoothsolutions of LIA which develops a singularity in the shape of a corner in finite time,see Fig. 1.

The initial value problem (10) arises in di!erent situations. It is a frictionlessmodel for the evolution after a line–line reconnection in the absence of counterflowfor superfluid 4He. See in particular Schwarz (1985) and Lipniacki (2000). And italso appears, this time for finite length filaments with appropriate boundary condi-tions, in some models for the fiber architecture of aortic valve leaflets (Peskin andMcQueen, 1994). In these three works numerical simulations of the correspondingproblems are given. In the first two the existence of friction allows the authors totreat the initial singularity of the filament, while in the latter paper, where no frictionis present, the initial kink is approximated by a hyperbola. As it was already men-tioned, the absence of friction makes it a completely integrable system and a solutionin closed form could be expected. This is the purpose of this paper. As a by-productwe are able to quantify some relevant parameters of the evolution. For example, inthe evolution after reconnection the kink located at the origin immediately smoothsout in a horseshoe-shaped curve around X#0, t$ with curvature

p , t > 0:

Here c0 is a constant determined by sin#!=2$ ! A1 ! e"#c20=2$": The trajectory X#0, t$ isa line and we call ’ the angle of that line with the plane which contains the initialkink, see Fig. 18 in Schwarz (1985) and Fig. 3 in Lipniacki (2000) (cf. with Fig. 1 inthis article). At the same time some wave phenomena appear along the two asymp-totic lines which decrease when friction is increased, see Fig. 4 in Lipniacki (2000).In this article we give in Theorem 1 an expression for ’ in terms of the curvature of

Figure 1. Kink formation. ! and ’ angles.

930 Gutierrez, Rivas, and Vega

Downloaded By: [University of Toronto] At: 16:23 2 December 2010

image of self-similar binormal curvature flow.

(from Gutiérrez, Rivas, and Vega, 2003)

definition of weak binormal curvature flow

Definition

Let I be an interval of R. A family of measures (Vt)t∈I on R3 × S2 is avery weak binormal curvature flow of finite mass if

1 Vt has no boundary for a.e.t in the sense that∫R3×S2

∇f (x) · ξVt(dx , dξ) = 0 for f ∈ C∞c (R3)

2 The map t 7→ Vt(R3 × S2) is finite and non-increasing on I.3 For every X ∈ D(R3, R3) the map t 7→

∫R3×S2 X · ξ Vt(dx , dξ) is

Lipschitz on I and for a.e. t ∈ I,

∫R3×S2

X · ξ Vt(dx , dξ) = −

∫R3×S2

D(curlX ) : (ξ⊗ ξ) Vt(dx , dξ).

remarks

Given any Lipschitz curve γ : S1 → R3, we can define anassociated measure Vγ by∫

R3×S2f (x , ξ)dVγ =

f (γ(s),γ ′(s)

|γ ′(s)|)|γ ′(s)| ds

If γ : I × S1 → R3 is a smooth solution, then the family ofassociated measures (Vt)t∈I is a weak solution.

the definition of weak solutions is linear with respect to V

weak flows may be useful for describing limits of sequences ofsolutions with rapidly oscillating tangents

it is easy to construct examples of weak solutions exhibitingchange of topology.

Existence and (failure of) uniqueness

for any lipschitz γ0 : S1 → R3, there exists a weak solution (Vt)t∈Rsuch that

wklimt→0 Vt = Vγ0

for Γt(X ) :=∫

S2 X (x) · ξVt(dx , dξ),

t 7→ Γt is continuous in suitable weak normsΓt is an integer multiplicity rectifiable 1-current for every t

Vt(R3 × S2) = |γ0| for all t .

The definition of weak solution only involves∫

S2 ξVt(·, dξ) and∫S2 ξ⊗ ξVt(·, dξ) — shocking failure of uniqueness

examples includeannihilating vortex rings — further failure of uniquenesspair of vortex rings born out of vacuum notcircle with “microscopic oscillations" —truly appalling failure ofuniqueness

for Γt(X ) :=∫

results: weak-strong uniqueness

Theorem (J-Smets, 2011)

Let γ ∈ C([0, T ], W 3,∞(S1, R3)) be a smooth binormal curvature flowwithout self-intersection.

Let(Vt)t∈[0,T ] be a weak binormal curvature flow.

If V0 = Vγ(0,·), then Vt = Vγ(t ,·) for all t in [0, T ].

1 proof also shows that a WBCF that starts near a smooth flowremains close for some time. This can be seen in numerics if we

believe that a simulation of (1) with rough data corresponds to a WBCF.

2 self-intersection should always lead to nonuniquenes.

3 theorem fails if t 7→ Vt(R3 × S2) is allowed to increase.

4 regularity here is much weaker than in Banica-Vega work.

results: an estimate for Schrödinger maps

Let u ∈ C(I, H3(T 1, S2)) be a solution of the Schrödinger map equation

ut = u × uss (3)

on I = (−T , T ) for some T > 0. Given any other solutionv ∈ L∞(I, H1/2(T 1, S2)) of (3), there exists a continuous functionσ : I → T 1 such that for every t ∈ I,

‖v(t , ·) − u(t , ·+ σ(t))‖L2(T 1, R3) 6 C‖v(0, ·) − u(0, ·)‖L2(T 1, R3),

where the constant C ≡ C(‖∂sssu(0, ·)‖L2 , T ), in particular C does notdepend on v .

Note that here there is no “non-self-intersection” condition.the critical Sobolev space is H1/2.

sharpness of the above

The above theorem is not true without the translation σ(t):

For any given t 6= 0, the flow map for (3) at time t is not continuous asa map from C∞(T 1, S2), equipped with the weak topology of H1/2, tothe space of distributions (C∞(T 1, R3))∗. Indeed, for any σ0 ∈ R thereexist a sequence of smooth initial data (um,σ0(0, ·))m∈N ∈ C∞(T 1, S2)

such that

um,σ0(0, ·) u∗(0, ·) in H12

weak(T1, R3),

where u∗(0, s) := (cos(s), sin(s), 0) is a stationary solution of (3), andfor any t ∈ R

um,σ0(t , ·) u∗(t , ·+ σ0t) in H12

weak(T1, R3).

key idea: relative tilt-excess

Given smooth curve γ(s), let rγ :=injectivity radius, and

P(x) := nearest point projection onto Image(γ),

when dist(x ,γ) < rγ;

Xγ(x) := f (dist(x ,γ(·)))γs(P(x)).

where f is smooth, 0 6 f 6 1, f ′ 6 0, and

f (r) := 1 − r2 for r 6 rγ/2, f (r) = 0 if r > rγ.

Finally, define

E(V ;γ) :=

∫R3×S2

(1 − Xγ(x) · ξ)V (dx , dξ)

LemmaIf V has no boundary and E(V ;γ) = 0, then V = cVγ for some c.

main estimate

Key Fact:Under the hypotheses of Weak-Strong Uniqueness Theorem,

E(Vt ;γt) 6 KE(Vt ;γt)

for almost every t ∈ [0, T ], where K is a constant depending only on rγand ‖γ‖L∞(I,W 3,∞(S1)).

If we know that V is parametrized by a single Lipschitz curve, we canmodify the definition of E , exploiting the parametrization to handleself-intersection of γ. A suitably modified version of the key fact stillholds and is used for the Schrödinger map “orbital continuity" theorem.

The proof of the key fact is a computation.....

.....a computation that goes like this:

Let ρ := 1 − (x − γ) · γss.

Then for any vector ξ, the following holds:

Xγ,t · ξ + D (curlXγ) : (ξ⊗ ξ)= (γss × (x − γ)) · (ξρ − γs)(γs · ξ)

+ (γss × γsρ ) · ξ (x − γ) · ξ

− f (d)(γsss × γs) · ξ(1 − γs·ξρ )

+ f (d)(γss · ξ)(x − γ) · (γsρ × γsss)

+ f (d) 1ρ2 ((γss × γs) · ξ) (γss · ξ)

+ f (d)γss×γsρ · ξγs·ξ

ρ (x − γ) · γsssf (d)

6 K (γ)(1 − Xγ · ξ) .

Here γs = γs P, etc.

Robert L. Jerrardcermics.enpc.fr/~al-hajm/ile-de-re/jerrard.pdfVery weak solutions of binormal...

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