Singularity formation in Nonlinear Evolution Equations Van Tien - PowerPoint PPT Presentation

Singularity formation in Nonlinear Evolution Equations Van Tien NGUYEN Workshop: Singular problems associated to quasilinear equations in celebration of Marie-Françoise Bidaut-Véron and Laurent Véron’s 70th Birthday June 2020 V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 1

Outline 1 Introduction Constructive approach 2 3 Results � Non-variational semilinear parabolic systems � Harmonic map heat flow + Wave maps � 2D Keller-Segel system Conclusion & Perspectives 4 V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 2

Introduction 1 - Introduction V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 3

Introduction Singularity formation in Nonlinear PDEs - Motivations � Applied point of view: - Understanding the physical limitation of mathematical models. Can the equations always do their job? What additional conditions of physical effects to have a proper model. - Singularities are physically relevant in natural sciences: concentration of laser beam in media (blowup in NLS), concentration of energy to smaller scales in fluid mechanics, concentration of density of bacteria population, etc. � Mathematical point of view: - The long-time dynamic of solutions to PDEs is of significant interest. However, solutions may develop singularities in finite time. How to extend solutions beyond their singularities? - The study of singularity formation requests new tools to handle many delicate problems such as stability of a family of special solutions, classification of all possible asymptotic behaviors , etc. - The numerical study of singularities is challenging. - ... V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 4

Introduction Model examples � Reaction-Diffusion equations: Non-variational semilinear parabolic systems  ∂ t u = ∆ u + v | v | p − 1 ,  µ > 0 , p , q > 1 . ( RD ) ∂ t v = µ ∆ v + u | u | q − 1 ,  Application: thermal reaction, chemical reaction, population dynamics, ... � Geometric evolution equations: Harmonic heat flows and wave maps ( σ -model): Φ( t ) : R d → S d , ∂ t Φ = ∆Φ + |∇ Φ | 2 Φ , ( HF ) t Φ = ∆Φ + � |∇ Φ | 2 − | ∂ t Φ | 2 � ∂ 2 Φ . ( WM ) Application: geometry, topology, simplified model for Einstein’s equation of general relativity,... � Aggregation-Diffusion equations: the 2D Keller-Segel system  ∂ t u = ∆ u − ∇ · ( u ∇ Φ u ) ,  in R 2 . ( KS ) 0 = ∆Φ u + u ,  Application: biology (chemotaxis), interacting many-particle system, ... V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 5

Introduction Framework of studying singularities in PDEs Blowup Numerical Numerical in PDEs methods methods Description Existence after blowup Obstructive Constructive Classification blowup set argument approach Stability rates & profiles less information Genericity asymptotic dynamics V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 6

Introduction Framework of studying singularities in PDEs Blowup Numerical Numerical in PDEs methods methods Description Existence after blowup Obstructive Constructive Classification blowup set argument approach Stability rates & profiles less information Genericity asymptotic dynamics V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 7

Constructive approach 2 - Constructive approach V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 8

Constructive approach Underlying problem Existence and Stability of blowup solutions. − − − − − − − − − − − − − − − − − − − − − − − − − � Obstructive argument (Virial law): Existence, no blowup dynamics. � Constructive approach : Existence + blowup dynamics. - Kenig (Chicago), Rodnianski (Princeton), Merle (Cergy Pontoise & IHES), Raphaël (Cambridge), Martel (École Polytechnique), Collot (CNRS & Cergy Pontoise), ... - del Pino (Bath), Musso (Bath), Wei (UBC), Davila (Bath), ... - Krieger (EPFL), Schlag (Yale), Tataru (Berkeley), ... - Herrero(UCM), Velázquez (Bonn), Zaag (CNRS & Paris Nord), ... V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 9

Constructive approach Architecture of the constructive approach � Constructing a good approximate solution; � Reduction of the linearized problem to a finite dimensional one: - Modulation technique: Kenig, Merle, Raphaël, Martel, ... � existence/stability; - Inner-outer gluing method: del Pino, Wei, Musso, Davila, ... � existence/stability; - Iterative technique: Krieger, Schlag, Tataru, ... � existence; - Spectral analysis � existence/stability + classification ; - ... � Solving the finite dimensional problem (if necessary). V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 10

Results Non-variational semilinear parabolic systems 3 - Results: The non-variational semilinear parabolic system  ∂ t u = ∆ u + v | v | p − 1 ,  µ > 0 , p , q > 1 . ( RD ) ∂ t v = µ ∆ v + u | u | q − 1 ,  Mathematical analysis: No variational structure � Energy-type methods break down; µ � = 1 � breaks any symmetry of the problem; The linearized operator is not self-adjoint even for the case µ = 1. Literature: Andreucci-Herrero-Velázquez ’97, Souplet ’09, Zaag ’98 & ’01, Mahmoudi- Souplet-Tayachi ’15, ... V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 11

Results Non-variational semilinear parabolic systems Type I (ODE-type) blowup solutions for ( RD ) via spectral analysis � Type I blowup: " ∂ t dominates ∆" � the blowup rate, unknown blowup profiles . � u ′ = ¯ � � � � v p , ¯ Γ( T − t ) − α , α = p + 1 pq − 1 , β = q + 1 ¯ u = � pq − 1 . v ′ = ¯ u q ¯ ¯ γ ( T − t ) − β v Theorem 1 (Ghoul-Ng.-Zaag ’18]) . � ∃ ( u 0 , v 0 ) ∈ L ∞ × L ∞ such that the solution ( u , v ) to System ( RD ) blows up in finite time T and admits the asymptotic dynamic ( T − t ) α u ( x , t ) − Φ 0 ( ξ ) → 0 , ( T − t ) β v ( x , t ) − Ψ 0 ( ξ ) → 0 , as t → T in L ∞ , where x √ • (blowup variable) ξ = ( T − t ) | ln( T − t ) | ; Ψ 0 ( ξ ) = γ (1 + b | ξ | 2 ) − β with b > 0. Φ 0 ( ξ ) = Γ(1 + b | ξ | 2 ) − α , • (profiles) � The constructed solution is stable under perturbation of initial data. Remark: - Other profiles are possible, but they are suspected to be unstable. - The existence of Type II blowup solutions remains unknown. V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 12

Results Non-variational semilinear parabolic systems Constructive proof for ( RD ): approximate blowup solution � φ ( y , s ) = ( T − t ) α u ( x , t ) � Self-similar variables: y = √ T − t , s = − ln( T − t ) , x ψ ( y , s ) = ( T − t ) β v ( x , t ) , � ∂ s φ + 1 2 y . ∇ φ + αφ = ∆ φ + | ψ | p − 1 ψ, ∂ s ψ + 1 2 y . ∇ ψ + βψ = µ ∆ ψ + | φ | q − 1 φ, - Nonzero constant solutions: (Γ , γ ) α Γ = γ p , βγ = Γ q . � Linearizing: ( φ, ψ ) = (Γ , γ ) + (¯ φ, ¯ ψ ), � ¯ H + M �� ¯ � � φ φ = � + "nonlinear quadratic term"; ∂ s ¯ ¯ ψ ψ where H + M has two positive eigenvalues 1 and 1 2 , a zero eigenvalue and an infinite many discrete negative spectrum. V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 13

Results Non-variational semilinear parabolic systems Constructive proof for ( RD ): approximate blowup solution � ¯ � � � � � � � � � φ f 2 f 2 a 2 a 0 y 2 + � Null mode is dominant : ( y , s ) = θ 2 ( s ) = , , ¯ ψ g 2 g 2 b 2 b 0 � θ 2 ∼ − 1 θ ′ c θ 2 2 + O ( | θ 2 | 3 ) , 2 = ¯ ¯ c > 0 , cs . ¯ � Inner approximation: for | y | ≤ C , � � � � � � − | y | 2 a ′ Γ φ ξ = | y | x ( y , s ) ∼ 2 √ s = � � ψ γ b ′ s ( T − t ) | ln( T − t ) | 2 V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 14

Results Non-variational semilinear parabolic systems Constructive proof for ( RD ): approximate blowup solution � Shape of profiles: � � � � � � Φ 0 ( ξ ) + 1 Φ 1 φ ξ = | y | ( y , s ) = ( ξ ) + . . . , √ s , Ψ 0 Ψ 1 ψ s where − ξ − ξ 2Φ ′ 2Ψ ′ 0 − α Φ 0 + Ψ p 0 = 0 , 0 − β Ψ 0 + Φ q 0 = 0 . - Solving ODEs: Φ 0 (0) = Γ, Ψ 0 (0) = γ , Φ 0 ( ξ ) = Γ(1 + b | ξ | 2 ) − α , Ψ 0 ( ξ ) = γ (1 + b | ξ | 2 ) − β , b ∈ R . � Matching asymptotic expansions: � value of b = b ( p , q , µ ) > 0. � V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 15

Results Non-variational semilinear parabolic systems Constructive proof for ( RD ): Control of the remainder � Linearized problem: ( φ, ψ ) = (Φ 0 , Ψ 0 ) + (Λ , Υ), � � �� � � � � Λ Λ R 1 = H + M + V + + "quadratic term" . ( ⋆ ) ∂ s Υ Υ R 2 � Constructing for ( ⋆ ) a global in time solution (Λ , Υ) such that � Λ( s ) � L ∞ + � Υ( s ) � L ∞ − → 0 as s → + ∞ . V. T. Nguyen (NYUAD) Singularities in Nonlinear PDEs 16