Variational methods for effective dynamics Robert L. Jerrard Department of Mathematics University of Toronto Minischool on Variational Problems in Physics October 2-3, 2014 Fields Institute Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 1 / 15
effective dynamics means: Given a nonlinear evolution equation with small or large parameter one seeks a simple description of (at least some) solutions, where simple may mean: in terms of lower-dimensional objects relevance of the calculus of variations Γ -convergence is very often a source of inspiration Γ -convergence (with related estimates) is often an ingredient in proofs including for example for wave and Schrödinger equations Γ -convergence (with upgrades) can sometimes be the basis for proofs especially for gradient flows, cf lectures of Ambrosio In general: effective dynamics is largely a question of stability and calculus of variations is very relevant to stability. Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 2 / 15
effective dynamics means: Given a nonlinear evolution equation with small or large parameter one seeks a simple description of (at least some) solutions, where simple may mean: in terms of lower-dimensional objects relevance of the calculus of variations Γ -convergence is very often a source of inspiration Γ -convergence (with related estimates) is often an ingredient in proofs including for example for wave and Schrödinger equations Γ -convergence (with upgrades) can sometimes be the basis for proofs especially for gradient flows, cf lectures of Ambrosio In general: effective dynamics is largely a question of stability and calculus of variations is very relevant to stability. Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 2 / 15
general open problems 1 . Abstract framework for Γ -convergence and Hamiltonian systems. For gradient flows, Sandier-Serfaty ’04, Serfaty ’11 2 . For Hamiltonian systems (especially), Can one ever establish global-in-time results ? In particular, given a periodic solution of a limiting Hamiltonian system, can one find “nearby" periodic solutions for the approximating functional? Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 3 / 15
First example Exercise Assume that F ε : R 2 ∼ = C → R and that F ε → F in some topology. For which topologies is it true that solutions of the ODEs ˙ x ε = −∇ F ε ( x ε ) , x ε ( 0 ) = x 0 (1) ¨ x ε ( 0 ) = x 0 , ˙ x ε = −∇ F ε ( x ε ) , x ε ( 0 ) = v 0 (2) i ˙ x ε = −∇ F ε ( x ε ) , x ε ( 0 ) = x 0 (3) converge, as ε → 0, to solutions of the ε = 0 systems? Note: for (1), “energy" decreases along trajectories: d dt F ε ( x ε ) = −| ˙ x ε | 2 x ε | 2 + F ε ( x ε )] = 0. for (2), “energy" is conserved: d dt [ 1 2 | ˙ for (3), (a different) “energy" is conserved: d dt F ε ( x ε ) = 0. Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 4 / 15
Another example The above exercise is misleading in that it (probably) doesn’t use gradient structure doesn’t distinguish between different dynamics more illustrative : Let g : R 2 → R be a fixed smooth function, and define F ε ( x , y ) := g ( x , y ) + ε − p ( y − ε q sin ( x ε )) 2 for certain p , q > 0. Exercise Show that � g ( x , 0 ) if y = 0 Γ F ε → F 0 ( x , y ) := + ∞ if not Exercise For which values of p , q do solutions of various ODEs for F ε converge to solutions for F 0 ? Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 5 / 15
Note : in general Γ -convergence is far too weak to allow any conclusions about dynamics. Exercise Assume that f : R n → R is continuous, φ : R n → R is measurable and Z n -periodic, with inf φ = 0. Then for any p > 0, F ε ( x ) := f ( x ) + ε − p φ ( x Γ ε ) − → f . Exercise Assume that f : R n → R is positive. Let { x i } be a countable dense subset of R n , and define � if x ∈ ∪ ∞ i = 1 B ( x i , 2 − i ε ) 0 F ε ( x ) := f ( x ) if not Then Γ F ε → F > 0 a.e. and in L 1 loc , but F ε ( x ) − → 0 . Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 6 / 15
For the duration of this lecture, we consider the Allen-Cahn energy � ε 2 |∇ u | 2 + 1 2 ε ( u 2 − 1 ) 2 dx u ∈ H 1 F ε ( u ) := loc (Ω) Ω and associated evolution problems (where typically Ω = R n .) Plan recall Γ -convergence (for inspiration) and state corresponding wave equation result discuss proof transform via change of variables into a stability question address this using variational arguments Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 7 / 15
Theorem (Modica-Mortola ’77, Modica ’87,Sternberg ’88) 1. (compactness) If ( u ε ) ε ∈ ( 0 , 1 ] is a sequence in H 1 (Ω) such that F ε ( u ε ) ≤ C then there is a subsequence that converges in L 1 as ε → 0 to a limit u ∈ BV (Ω; {± 1 } ) . L 1 2. (lower bound) If ( u ε ) ⊂ H 1 (Ω) and u ε → u, then � 4 3 | Du | (Ω) if u ∈ BV (Ω; {± 1 } ) lim inf ε → 0 F ε ( u ε ) ≥ F 0 ( u ) := + ∞ if not 3. (upper bound) For any u ∈ L 1 (Ω) there exists a sequence ( u ε ) ⊂ H 1 (Ω) such that L 1 u ε → u and lim sup F ε ( u ε ) ≤ F 0 ( u ) . ε → 0 Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 8 / 15
Informally, F ε ( · ) Γ → “ interfacial area functional ” As a corollary : if ( u ε ) is a sequence of minimizers of F ε (for suitable boundary data....) then u ε → u ∈ BV (Ω; {± 1 } ) after passing to a subsequence if necessary , and the set { x ∈ Ω : u ( x ) = 1 } has minimal perimeter in Ω . Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 9 / 15
further (PDE) results : ( many references omitted here.....) similar for nonminimizing solutions of − ε ∆ u ε + 1 ε ( u 2 ε − 1 ) u ε = 0 (assuming natural energy bounds.) solutions of − ε ∆ u ε + 1 ε ( u 2 ε − 1 ) u ε = ε κ are related in a similar way to surfaces of Constant Mean Curvature. in addition, u ε ( x ) ≈ q ( d ( x ) ) , where d ( · ) is signed distance from interface, ε |∇ d | 2 = 1 , so that d ( · ) satisfies d = 0 on Γ . Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 10 / 15
Theorem (J ’11, Galvão-Sousa and J., ’13) Assume that Γ is a smooth, compact, embedded, timelike hypersurface in ( T ∗ , T ∗ ) × R n , bounding a set O , and such that H mink (Γ) = κ ∈ R . Then there exists a sequence of solutions ( u ε ) of the wave equation ε ( ∂ tt u ε − ∆ u ε ) + 1 ε ( u 2 − 1 )( 2 u − εκ ) = 0 such that � 1 in O in L 2 loc (( T ∗ , T ∗ ) × R n ) u ε → u := in O c − 1 In fact we prove more, including energy concentration around Γ , estimates of rate of convergence etc. H mink = ( 1 − v 2 ) − 1 / 2 ( H euc − ( 1 − v 2 ) − 1 a ) , where v = velocity , a = acceleration . H mink = 0 ⇐ ⇒ critical point of Minkowskian area functional Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 11 / 15
Formal arguments and elliptic results suggest that u ≈ q ( d ε ) , where q is the optimal 1-d profile: − q ′′ + ( q 2 − 1 ) q = 0 , q ( 0 ) = 0 , q → ± 1 at ± ∞ d is the signed Minkowski distance function to Γ , i.e. − ( ∂ t d ) 2 + |∇ d | 2 = 1 , d = 0 on Γ , Note that q minimizes � 2 ( v ′ ) 2 + 1 1 2 ( 1 − v 2 ) 2 dr v �→ R among functions such that v ( r ) → ± 1 as r → ±∞ . Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 12 / 15
Plan: consider 1 d case (with κ � = 0) for simplicity Let r 0 = κ − 1 . Then Γ = { r 0 ( sinh θ, cosh θ ) : θ ∈ R } := { ( t , x ) : x 2 − t 2 = r 2 0 } change to Minkowskian polar coordinates ( r cosh θ, r sinh θ ) = ( x , t ) , r > 0 , θ ∈ R Then θ ≈ “ time ” and r − r 0 = d = Minkowski distance to Γ then hope to show that u ( x , t ) = v ( r , θ ) ≈ q ( d ε ) = q ( r − r 0 ) . ε in fact we will concoct a functional ζ such that η ( v ( · , θ )) small ⇒ v ≈ q ( r − r 0 ) , ε d θη ( v ( · , θ )) ≈ 0 if v ≈ q ( r − r 0 d ) . ε Robert L. Jerrard (Toronto ) Variational methods for effective dynamics Variational Problems in Physics 13 / 15
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