The method of concentration compactness and dispersive Hamiltonian - PowerPoint PPT Presentation

The method of concentration compactness and dispersive Hamiltonian Evolution Equations W. Schlag, http://www.math.uchicago.edu/˜schlag Aalborg, August 2012 W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Overview Goal: To describe recent advances in large data results for nonlinear wave equations � u = F ( u , Du ) , F ( 0 ) = DF ( 0 ) = 0 , ( u ( 0 ) , ˙ u ( 0 )) = ( f , g ) Small data theory: F treated as perturbation. Local/Global well-posedness, conserved quantities (energy), symmetries (especially dilation), choice of spaces, algebraic properties of F (nullforms) Large data: local-in-time existence, energy subcritical problems: time of existence depends on energy of data, so can time-step. Problem: no information on long-term dynamics such as scattering (solutions are asymptotically free). Finite-time breakdown (blowup) of solutions may occur (type I and II). Classification of possible blowup dynamics Induction of energy to prove scattering for global solution: If false then there exists a minimal energy E ∗ where it fails. Construct critical solution ( minimal criminal ) u ∗ with energy E ∗ . W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Overview u ∗ enjoys compactness properties modulo symmetries. Forward trajectory ( u ∗ ( t ) , ∂ t u ∗ ( t ) ), t ≥ 0 pre-compact in energy space. Idea: if not compact, then by the method of concentration compactness u ∗ decomposes into different solutions with strictly smaller energies than E ∗ . By induction hypothesis, each of these solutions has the desired property and by means of suitable perturbation theory one shows that u ∗ then also possess this property. Rigidity: Show that u ∗ with this property cannot exist. Kenig-Merle scheme Concentration compactness much more versatile, is not tied to induction on energy: key ingredient in the classification of blow-up behavior. W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Calculus of Variations Sobolev imbedding in R 3 : � f � L p ( R 3 ) ≤ C � f � H 1 ( R 3 ) , 2 < p < 6 What are the extremizers, optimal constant? Variational problem: � � � � � f � L p ( R 3 ) = 1 inf � f � H 1 ( R 3 ) = µ > 0 � Minimizing sequence { f n } ∞ n = 1 ⊂ H 1 ( R 3 ) , � f n � p = 1 , � f n � H 1 ( R 3 ) → µ How to pass to a limit f n → f ∞ strongly in L p ( R 3 ) ? Loss of compactness due to translation invariance! n = 1 ⊂ R 3 such that Claim for p < 6: there exists a sequence { y n } ∞ { f n ( · − y n ) } ∞ n = 1 precompact in L p ( R 3 ) and H 1 ( R 3 ) . W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Loss of compactness Figure: masses separating Simplified model: Assume that f n = g n + h n where � g n � p p = m 1 > 0 and � h n � p p = m 2 > 0, m 1 + m 2 = 1, supports of g n , h n disjoint. H 1 ≥ µ 2 ( m 2 / p + m 2 / p Then � f n � 2 H 1 = � g n � 2 H 1 + � h n � 2 ) , 2 / p < 1 1 2 This is a contradiction since right-hand side > µ 2 . W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

A concentration-compactness decomposition { f n } ∞ n = 1 ⊂ H 1 ( R 3 ) a bounded sequence. Then ∀ j ≥ 1 there ∃ (up to n = 1 ⊂ R 3 and V j ∈ H 1 such that subsequence) { x j n } ∞ j = 1 V j ( · − x j for all J ≥ 1 one has f n = � J n ) + w J n ∀ j � k one has | x j n − x k n | → ∞ as n → ∞ n ( · + x j w J n ) ⇀ 0 for each 1 ≤ j ≤ J as n → ∞ lim sup n →∞ � w J n � L p ( R 3 ) → 0 as J → ∞ for all 2 < p < 6 Moreover, as n → ∞ , 2 = � J � f n � 2 j = 1 � V j � 2 2 + � w J n � 2 2 + o ( 1 ) �∇ f n � 2 2 = � J j = 1 �∇ V j � 2 2 + �∇ w J n � 2 2 + o ( 1 ) P . G´ erard 1998, more explicit form of P . L. Lions’ concentration-compactness trichotomy for measures. Makes failure of compactness modulo symmetries explicit. immediately implies compactness claim for minimizing sequences: V j = 0 for j > 1. only noncompact symmetry groups matter (no rotations)! W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

The profiles V j in the L p sea We fish for more profiles from the sea: w 3 n ( · + y n ) ⇀ V 4 W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Euler-Lagrange equation Pass to limit f n ( · − y n ) → f ∞ in H 1 ( R 3 ) , � f ∞ � p = 1, � f ∞ � H 1 = µ . Can assume f ∞ ≥ 0. Then ∃ λ > 0 Lagrange multiplier − ∆ f ∞ + f ∞ = λ | f ∞ | p − 2 f ∞ Remove λ > 0 since p > 2. Then f ∞ = Q > 0 solves − ∆ Q + Q = | Q | p − 2 Q ( ∗ ) Q ∈ H 1 , Q > 0 unique up to translation (Kwong 1989, McLeod 93). Q is exponentially decaying, radial, smooth. For dim = 1 explicit formula, only solutions to ( ∗ ) in H 1 ( R ) are 0 , ± Q . For d > 1 have infinitely many radial solutions to ( ∗ ) that change sign (nodal solutions). Berestycki, Lions, 1983. W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

What happens for p = 6? Decomposition from above fails at p = 6 due to dilation symmetry. Correct setting is ˙ H 1 ( R 3 ) since � f � L 6 ( R 3 ) ≤ C � f � ˙ H 1 ( R 3 ) = C �∇ f � 2 ( † ) Translation and scaling invariant, noncompact group actions. n = 1 ⊂ ˙ { f n } ∞ H 1 ( R 3 ) a bounded sequence. Then ∀ j ≥ 1 there ∃ (up to n = 1 ∈ R + and V j ∈ ˙ H 1 such that subsequence) { x j n = 1 ⊂ R 3 , { λ j n } ∞ n } ∞ � λ j n V j ( λ j n ( · − x j for all J ≥ 1 one has f n = � J n )) + w J j = 1 n ∀ j � k one has λ j n + λ k n + λ j n | x j n − x k n | → ∞ as n → ∞ λ k λ j lim sup n →∞ � w J n � L 6 ( R 3 ) → 0 as J → ∞ . Moreover, as n → ∞ , J � �∇ f n � 2 �∇ V j � 2 2 + �∇ w J n � 2 2 = 2 + o ( 1 ) j = 1 W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Minimizer for p = 6 Variational problem associated with ( † ) � � � � � f � L 6 ( R 3 ) = 1 = µ > 0 inf � f � ˙ � H 1 ( R 3 ) Minimizing sequence n = 1 ⊂ ˙ { f n } ∞ H 1 ( R 3 ) , � f n � L 6 ( R 3 ) = 1 , � f n � ˙ H 1 ( R 3 ) → µ From the decomposition/minimization: Exactly one profile n = 1 ∈ R + such that { λ 1 / 2 ∃{ y n } ∞ n = 1 ⊂ R 3 , { λ n } ∞ n f n ( λ n ( · − y n )) } ∞ n = 1 precompact in L 6 ( R 3 ) and ˙ H 1 ( R 3 ) . λ 1 / 2 n f n ( λ n ( · − y n )) → f ∞ , Euler-Lagrange equation for ϕ = cf ∞ ∆ ϕ + ϕ 5 = 0 Only radial solutions are ± W , 0 up to dilation symmetry, where W ( x ) = ( 1 + | x | 2 / 3 ) − 1 2 W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Calculus of Variations on Minkowski background Let � 1 � − u 2 t + |∇ u | 2 � L ( u , ∂ t u ) := ( t , x ) dtdx (1) 2 R 1 + d t , x Substitute u = u 0 + ε v . Then � ( � u 0 )( t , x ) v ( t , x ) dtdx + O ( ε 2 ) L ( u , ∂ t u ) = L 0 + ε R 1 + d t , x where � = ∂ tt − ∆ . Thus u 0 is a critical point of L if and only if � u 0 = 0. Significance: Underlying symmetries ⇒ invariances ⇒ Conservation laws Conservation of energy, momentum, angular momentum Lagrangian formulation has a universal character, and is flexible, versatile. W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Wave maps 1 Let ( M , g ) be a Riemannian manifold, and u : R 1 + d → M smooth. t , x What does it mean for u to satisfy a wave equation? Lagrangian d � 1 � � 2 ( −| ∂ t u | 2 | ∂ j u | 2 L ( u , ∂ t u ) = g + dtdx g R 1 + d j = 1 t , x Critical points L ′ ( u , ∂ t u ) = 0 satisfy “manifold-valued wave equation”. M ⊂ R N imbedded, this equation is � u ⊥ T u M or � u = A ( u )( ∂ u , ∂ u ) , A being the second fundamental form. For example, M = S n − 1 , then � u = u ( | ∂ t u | 2 − |∇ u | 2 ) Note: Nonlinear wave equation, null-form! Harmonic maps are solutions. W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness

Wave maps 2 Intrinsic formulation: D α ∂ α u = η αβ D β ∂ α u = 0, in coordinates tt + ∆ u i + Γ i jk ( u ) ∂ α u j ∂ α u k = 0 − u i η = ( − 1 , 1 , 1 , . . . , 1 ) Minkowski metric Similarity with geodesic equation: u = γ ◦ ϕ is a wave map provided � ϕ = 0, γ a geodesic. � � � | ∂ t u | 2 g + � d j = 1 | ∂ j u | 2 Energy conservation: E ( u , ∂ t u ) = dx g R d is conserved in time. Cauchy problem: � u = A ( u )( ∂ α u , ∂ α u ) , ( u ( 0 ) , ∂ t u ( 0 )) = ( u 0 , u 1 ) smooth data. Does there exist a smooth local or global-in-time solution? Local: Yes. Global: depends on the dimension of Minkowski space and the geometry of the target. W. Schlag, http://www.math.uchicago.edu/˜schlag Concentration Compactness