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Remarks on PDE's

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Remarks on PDE's Powered By Docstoc
					Sergiu Klainerman




        Great Problems in
        Nonlinear Evolution
            Equations
                    On the Analysis of Geometric
                        Evolution Equations


                                         Los Angeles, August 2000
   GOOD PROBLEMS ACCORDING TO HILBERT
1. Clear and easy to comprehend

2. Difficult yet not completely inaccessible
        “A mathematical problem should be difficult, in order to entice us, yet not
        completely inaccessible lest it mocks at our efforts. It should provide a landmark on
        our way through the confusing maze and thus guide us towards hidden truth.”
 Should lead to meaningful                             Should be related to meaningful
 generalizations                                       simpler problems
  If we do not succeed in solving a mathematical        ``In dealing with mathematical problems,
  problem , the reason is often do to our failure to    specialization plays, I believe, a still more
  recognize the more general standpoint from            important part then generalization. Perhaps
  which the problem before us appears only as a         in most cases where we seek in vain the
  single link in a chain of related problems’’          answer to a question, the cause of the
                                                        failure lies in the fact that problems simpler
                                                        than the one in hand have been either not
                                                        at all or incompletely solved.’’




 3. Should provide a strategic height towards a broader goal
TABLE OF CONTENTS

 1. PDE AS A UNIFIED SUBJECT


 2. REGULARITY OR BREAK-DOWN


 3. MAIN GOALS


 4. MAIN OPEN PROBLEMS


 5. RELATED OPEN PROBLEMS
             PDE AS A UNIFIED SUBJECT


                  How to generate       interesting PDEs




                            n                                              n+1
Euclidian Space:         R                 Minkowski Space: R

            = 12 + … + n2                         = - t2 + 12 + … + n2


Riemannian              (M,g)                Lorentzian                (M,g)
                          Simplest differential
    g= g i j  i  j    operators invariant under
                            the isometry group       g=   g   
             PDE AS A UNIFIED SUBJECT


               How to generate       interesting PDEs



Symmetries                     Conservation Laws


Geometric Variational               Euler-Lagrange
           Variational
Lagrangian   principle                equations
           principle

              Well-defined Limits

              Symmetry Reductions                       Effective
              Phenomenological Reductions
                                                        equations
                PDE AS A UNIFIED SUBJECT


1.Fundamental Laws                       • Geometric
  Obtained from a simple geometric         (Elliptic)
  Lagrangian                             • Mathematical Physics
                                            (Hyperbolic)

2.Effective Equations                    • Well-defined Limits
  Derived from the fundamental             Newtonian limit
  equations by taking limits or making     Incompressible limit
  specific simplifying assumptions       • Symmetry Reduction
                                         • Phenomenological

3.Diffusive Equations                    • Geometric
                                           Heat flows
                                         • Mathematical Physics
                                           Stochastic
4. Others
                PDE AS A UNIFIED SUBJECT


1. Fundamental Laws                    Geometrical Equations Elliptic
 Obtained from a simple geometric            •Cauchy- Riemann
 Lagrangian                                  • Laplace
  Equations which play a fundamental
  role in Mathematics . Find objects          • Dirac
  with optimal geometric properties.          • Hodge systems
                                              • Harmonic Maps
2. Effective Equations                        • Yang- Mills
                                              •Ginsburg-Landau
                                              • Seiberg-Witten

3. Diffusive Equations                        • Minimal Surfaces
                                              • Einstein metrics


4. Other
             PDE AS A UNIFIED SUBJECT


1. Fundamental Laws                  Physical Equations   Hyperbolic
  Obtained from a simple geometric
                                         Relativistic Field Theories:
  Lagrangian
                                             • Wave and Klein-Gordon
  Equations which correspond to              equations
  our major physical theories                • Maxwell
                                             • Wave Maps
                                             • Yang Mills
2. Effective Equations                       • Einstein Field equations

                                         Relativistic Continuum
                                         Mechanics
3. Diffusive Equations                       • Elasticity
                                             • Gas dynamics
                                             • Magneto fluid-dynamics
4. Other
            PDE AS A UNIFIED SUBJECT
               OUR MAIN EQUATIONS

• Scalars
       Nonlinear Klein-Gordon
• Connections on a Principal Bundle
       Yang-Mills
• Lorentzian or Riemannian metrics

       Einstein equations
• Mappings between Manifolds

       Harmonic and Wave Maps
       Elasticity, Hydrodynamics, MHD
       Minimal Surface Equation

• Composite Equations
            PDE AS A UNIFIED SUBJECT

                                        • Well-defined Limits
1.Fundamental Laws                          Newtonian limit
                                             (non-relativistic)
                                                •Schrödinger
                                                • Elasticity
                                                • Gas dynamics

2. Effective Equations                      Incompressible limit
 Derived from the fundamental               Euler equations
 equations by taking limits or making
 specific simplifying assumptions       • Symmetry Reductions
                                                • stationary
                                                • spherically symmetric
                                                • dimensional reduction
3. Diffusive Equations                  • Phenomenological
                                            •Dispersive( KdV,Schrödinger)
                                            • Ginsburg-Landau
4. Other                                    • Maxwell-Vlasov
 PDE AS A UNIFIED SUBJECT


1.Fundamental Laws
                         Parabolic

                         • Geometrical Equations
                             • Ricci Flow
2. Effective Equations       • Harmonic Map Flow
                             • Gauss Flow
                             • Mean Curvature Flow
                             • Inverse Mean Curvature Flow

                         • Physical Equations
3. Diffusive Equations       Macroscopic limit
                             • Compressible Fluids (heat
                               conduction)
                             • Navier-Stokes (viscosity)
                             • Electrodynamics (resistivity)
4. Other
              PDE AS A UNIFIED SUBJECT
             CONSERVATION LAWS, A-PRIORI BOUNDS

Noether Theorem:
                                                            Energy,
To any symmetry of the Lagrangean there                     Linear Momentum
corresponds a Conservation Law.                             Angular Momentum,
                                                            Charge


The basic physical equations have a limited number Conservation Laws.
The Energy provides the only useful, local, a-priori estimate.

 Are there other stronger a-priori bounds

Symmetry reductions generate additional Conservation Laws • Integrable Systems
                                                             • 2-D Fluids


Elliptic and diffusive equations possess additional a-priori •Maximum principle
                                                             • Monotonicity
estimates.
          REGULARITY OR BREAK-DOWN



                      Solutions to our basic nonlinear
                      equations, corresponding to
                      smooth initial conditions, may
                      form singularities in finite time,
                                                            Why?
  What is the
  character of        despite the presence of
the singularities?    conserved quantities .
                                                            Where?




           Can solution be continued                When?
             past the singularities?
        REGULARITY OR BREAK-DOWN



The problem of possible break-down of solutions to
interesting, non-linear, geometric and physical systems
is:
              the most basic               the most conspicuous unifying
              problem in PDE               problem; it affects all PDE




Intimately tied to the basic mathematical question of understanding
what we actually mean by solutions and, from a physical point of
view, to the issue of understanding the very limits of validity of the
corresponding physical theories.
 REGULARITY OR BREAK-DOWN



 SUBCRITICAL    A-PRIORI BOUNDS(ENERGY)
    E>N               (E=strength of the bound )


  CRITICAL
   E=N

                                       SCALING
                       (N=strength of nonlinearity)
SUPERCRITICAL
    E<N
    REGULARITY OR BREAK-DOWN

                                           GENERAL
                                       EXPECTATIONS

 SUBCRITICAL
    E>N     Expect global regularity for all data.




   CRITICAL
    E=N           Expect, in most cases, global regularity
                  for all data.



SUPERCRITICAL
    E<N     Expect global       regularity for “small''
                   data. Expect large data breakdown.
  REGULARITY OR BREAK-DOWN

                                      GENERAL
                                  EXPECTATIONS
 SUBCRITICAL
    E>N
                     What is the character of the
                            breakdown?
   CRITICAL
                          Can solutions be
    E=N                     extended past
                          the singularities?



SUPERCRITICAL
    E<N     Expect global   regularity for “small''
               data. Expect large data breakdown.
                MAIN GOALS


1 To understand the problem of evolution for the
   basic equations of Mathematical Physics.



2 To  understand in a rigorous mathematical fashion
   the range of validity of various approximations.



3 To devise and analyze the right equation as a tool
   in the study of the specific geometric or physical
   problem at hand.
                   MAIN GOALS

1  To understand the problem of evolution for the
basic equations of Mathematical Physics.

    • Provide mathematical justification to the
    classification between sub-critical, critical and
    super-critical equations.

    • Determine when and how classical(smooth)
    solutions to our main supercritical equations
    form singularities.

    • Find an appropriate notion of global, unique      CAUSALITY
    solutions, corresponding to all reasonable
    initial conditions.

    • Determine the main asymptotic features of
    the general solutions.
                        MAIN GOALS

2  To understand in a rigorous mathematical fashion
the range of validity of various approximations.

    • Newtonian limit                   speed of light 
    • Incompressible limit            speed of sound 
    • Macroscopic limit            number of particles 
    • Inviscid limit                 Reynolds number. 


The dynamics of effective equations may lead to behavior which
is incompatible with the assumptions made in their derivation.



 Should we continue to trust     Should we abandon them in
and study them, nevertheless,    favor of the original equations
for pure mathematical reasons?   or a better approximation?
                        MAIN GOALS

3 To devise and analyze the right equation as a tool in
the study of a specific geometric or physical problem.
CALCULUS OF VARIATIONS            EVOLUTION OF EQUATIONS

Geometrical Equation Elliptic     Geometric Flows   Parabolic
  •Cauchy- Riemann
  • Laplace                          • Ricci
                                     •Harmonic Map
   • Dirac
                                     • Gauss
   • Hodge systems
                                     • Mean Curvature
   • Harmonic Maps
                                     • Inverse Mean Curvature
   • Yang- Mills
   • Saiberg-Witten
   • Minimal Surfaces
   • Einstein metrics
                       MAIN GOALS

3. To devise and analyze the right equation as a tool in
the study of a specific geometric or physical problem.
To be able to handle its
                                   EVOLUTION OF EQUATIONS
solutions past possible
singularities. To find a useful    Geometric Flows Parabolic
concept of generalized                • Ricci
solutions.                            •Harmonic Map
                                       • Gauss
 Penrose inequality using the
                                       • Mean Curvature
 inverse mean curvature flow.
                                       • Inverse Mean Curvature

 Results in 3-D and 4-D Differential
 Geometry using the Ricci flow.
 Attempt to prove the Poincare and
 geometrization conjecture .
          MAIN OPEN PROBLEMS


1   Cosmic Censorship in General Relativity

2   Break-down for 3-D Euler Equations


3   Global Regularity for Navier-Stokes


4   Global Regularity for other Supercritical Equations


5   Global Singular Solutions for 3-D Systems of
    Conservation Laws
                     MAIN OPEN PROBLEMS
1   Cosmic Censorship in General Relativity
                         EINSTEIN VACUUM EQUATIONS
          (M,g)               R -1/2 R g =0

     Initial Data Sets                            Asymptotic Flatness




    Cauchy
    Development
                  MAIN OPEN PROBLEMS
1   Cosmic Censorship in General Relativity

 Known Results

 Existence and Uniqueness
                                Any (, g, k) has a unique, future,
(BRUHAT-GEROCH)
                                Maximal Cauchy Development (MCD).
                                It may not be geodesically complete.

 Singularity Theorem            The future MCD of an initial data set
(PENROSE)                       (, g, k ) which admits a trapped
                                surface is geodesically incomplete.


Global Stability of Minkowski The MCD of an AF initial data set
(CHRISTODOULOU-KLAINERMAN)
                              (, g, k) which verifies a global
                              smallness assumption is geodesically
                              complete. Space-time becomes flat in
                              all directions.
                   MAIN OPEN PROBLEMS
1   Cosmic Censorship in General Relativity

Weak Cosmic Censorship

      Generic S.A.F. initial data sets have maximal, future, Cauchy
      developments with a complete future null infinity. All singularities
      are covered by Black Holes.
                    Naked singularities are non-generic

Strong Cosmic Censorship
      Generic S.A.F. initial data sets have maximal future Cauchy
      developments which are locally in-extendible as Lorentzian
      manifolds. Curvature singularities


                             Solutions are either geodesically
                             complete or, if incomplete,
                             end up in curvature singularities.
                MAIN OPEN PROBLEMS
1   Cosmic Censorship in General Relativity




                                      1       2   3   4
                     MAIN OPEN PROBLEMS
1   Cosmic Censorship in General Relativity

Known Results
                                   • Formation of trapped surfaces
Spherically Symmetric-Scalar       • Sharp smallness assumption
Field Model                        (implies complete regular solutions).
(D. Christodoulou)                 Scale invariant BV space
                                   • Examples of solutions with naked
                                   singularities
                                   • Rigorous proof of the weak and
                                   strong Cosmic Censorship

Results for U(1)U(1) symmetries
and Bianchi type
                     MAIN OPEN PROBLEMS
2   Break-down for 3D Euler Equations

     t u+u  u= -p         RR3           Initial Data (regular)
     div u= 0                                u(0, x)=u0 ( x )

    Known Results

    Local in time existence      For any smooth initial data there exists
                                 a T>0 and a unique solution in [0,T]R3.

    Continuation Theorem         The solution can be smoothly
    (BEALS-KATO-MAJDA)           continued as long as the vorticity 
                                 remains uniformly bounded; in fact as
                                 long as



                                                     Vorticity =u
                      MAIN OPEN PROBLEMS
2   Break-down for 3D Euler Equations                     t u+u  u= -p
                                                                 div u= 0
                            Most unstable equation.

    CONJECTURE

    Weak Form                                 Strong Form
    There exists:                             Most regular data lead to such
        • a regular data u0,                  behavior. More precisely the set
        • a time T* = T* (u0 )> 0             of initial data which lead to finite
        • a smooth uC∞( [0, T* ) R3 )       time break-down is dense in the
     ||(t)||L        as t  T* .          set of all regular data with
                                              respect to a reasonable topology.


    There may in fact exist plenty of global smooth solutions which are,
    however, unstable. More precisely the set of all smooth initial data which
    lead to global in time smooth solutions may have measure zero, yet, it may be
    dense in the set of all regular initial conditions, relative to a reasonable
    topology.
                MAIN OPEN PROBLEMS

3   Global Regularity for Navier-Stokes
 t u+u  u- u = -p      RR3              Initial Data (regular)
 div u= 0                                                u(0,x)=u0 (x)

Known Results

Local in time existence   For any smooth initial data there exists
                          a T>0 and a unique solution in [0,T]R3.

Continuation Theorem      The solution can be smoothly continued
(SERRIN)                  as long as the velocity u remains
                          uniformly bounded; in fact as long as
                  MAIN OPEN PROBLEMS

3   Global Regularity for Navier-Stokes
                                           t u+u  u- u = -p
                                                        div u= 0
CONJECTURE
The solutions corresponding to
                                           NOTE OF CAUTION
all regular initial data can be
smoothly continued for all t≥0.         Break-down requires infinite
                                        velocities--unphysical:
It is however entirely possible that    • incompatible with relativity
singular solutions exist but are        • thin regions of infinite velocities
unstable and therefore difficult to     are incompatible with the
construct analytically and impossible   assumption of small mean free
to detect numerically.                  path required in the macroscopic
                                        derivation of the equations.
The solutions corresponding to
generic, regular initial data can be
continued for all t≥0.
                    MAIN OPEN PROBLEMS
4    Global Regularity for other Supercritical Equations

          - V’() = 0,    RR3                      Initial Data at t=0
                                                       =f,     t  = g
                 V =  p+1


                                                      Known Facts

    Subcritical      p<5
                                  Global regularity for all data
    Critical         p=5
                             For any smooth initial data there exists
                             a T>0 and a unique solution in [0,T]R3.
    Supercritical    p>5     The solution can be smoothly continued
                             as long as
                 MAIN OPEN PROBLEMS
4   Global Regularity for other Supercritical Equations

       - V’() = 0,    RR3
              V =  p+1

                               Numerical results suggest
                               global regularity for all data.


      Subcritical     p<5
                                CONJECTURE
      Critical        p=5
                                There exist unstable solutions
      Supercritical   p>5
                                which break down in finite time.
                                Global Regularity for all
                                generic data.
                 MAIN OPEN PROBLEMS
5   Global, Singular, Solutions for the 3-D Systems of
    Conservation Laws

         u = (u1, u2 ,…, uN) ;      F0, F1, F2 , F3 : RN  RN

         u = u(t , x1, x2, x3 )

                        t F0(u)+∑3i =1 i Fi(u )= 0


     Compressible Euler Equations-ideal gases


                             Nonlinear Elasticity-hyperelastic materials
                     MAIN OPEN PROBLEMS
5    Global, Singular, Solutions for the 3-D Systems of
     Conservation Laws



    Known Results

    Local in time existance     For any smooth initial data there exists
                                a T>0 and a unique solution in [0,T]R3.

    Singularities               There exist arbitrarily small
    (JOHN, SIDERIS)             perturbations of the trivial data set
                                which break-down in finite time


    1-D Global Existence         Global existence and uniqueness for
    and Uniqueness              all initial data with small bounded
    (GLIMM, BRESSAN-LIU-YANG)   variation.
                   MAIN OPEN PROBLEMS
5   Global, Singular, Solutions for the 3-D Systems of
    Conservation Laws

     Find an appropriate                 NOTE OF CAUTION
    concept of generalized
    solution, compatible with     A full treatment of the Compressible
                                  Euler equations must include the
    shock waves and other
                                  limiting case of the incompressible
    possible singularities, for
                                  equations. This requires not only to
    which we can prove            settle the break-down conjecture 2 but
    global existence and          also a way of continuing the solutions
    uniqueness of the initial     past singularities.
    value problem.
     For generic data ?           Need to work on vastly simplified model
                                  problems.
                  MAIN OPEN PROBLEMS

CONCLUSIONS
I. All five problems seem inaccessible at the present time

II. Though each problem is different and would ultimately
require the development of custom-tailored techniques they
share important common characteristics.
    • They are all supercritical                       The development of such
    • They all seem to require the development of    methods may be viewed as
    generic methods which allow the presence of      one of the great challenges
    exceptional sets of data.                               for the next century.
    •Problems 1,4,5 require the development of a
    powerful hyperbolic theory comparable with the
    progress made last century in elliptic theory.

III. Need to concentrate on simplified model problems
  There are plenty of great simplified model problems in
  connection with Cosmic Censorship. Also problems 4 and 5.
  Problems 2 and 3 seem irreducible hard !
               RELATED OPEN PROBLEMS

III. Need to concentrate on simplified model problems:

     1. Stability of Kerr

     2. Global Regularity of Space-times with U(1) symmetry

     3. Global regularity of the Wave Maps from R2+1 to H2

     4. Small energy implies regularity - Critical case

     5. Strong stability of the Minkowski space

     6 . Finite L2 - Curvature Conjecture

     7. Critical well-posedness for semi-linear equations
              RELATED OPEN PROBLEMS

III. Need to concentrate on simplified model problems:



   8. The problem of optimal well- posedness for nonlinear
      wave and hyperbolic equations

   9. Global Regularity for the Maxwell-Vlasov equations

   10. Global Regularity or break-down for the supercritical
       wave equation with spherical symmetry

   11. Global stability for Yang-Mills monopoles and
       Ginsburg-Landau vortices

   12. Regularity or Break-down for quasi-geostrophic flow
            RELATED OPEN PROBLEMS

1. Stability of Kerr                               Cosmic Censorship


 (M,g)           R -1/2 R g = 0

                            CONJECTURE
                            Any small perturbation of the initial data set
      Compatible            of a Kerr space-time has a global future
       with weak            development which behaves asymptotically
    cosmic censorship       like (another) Kerr solution.



 Are Kerr solutions unique among all stationary solutions ? ELLIPTIC

 Do solutions to the linear wave equation on a Kerr (Schwartzschild)
 background decay outside the event horizon ? At what rate ?
              RELATED OPEN PROBLEMS

2. Global Regularity of                         Cosmic Censorship
   Space-times with U(1)
   symmetry
                                                  (M,g, )
2+1 Einstein equations coupled with a
wave map with target the hyperbolic
                                                R =  
space H2.                                         =0

                         Critical !              polarized U(1)


CONJECTURE
All asymptotically flat U(1) solutions of the
Einstein Vacuum Equations are complete.
                RELATED OPEN PROBLEMS

  3. Global regularity of
                                                     Cosmic Censorship
      the 2+1 Wave Maps
      to hyperbolic space.

 : IR 2+1              IH2
       I                                               (0)=f,     t (0) = g
      +     IJK   ()        J
                                      K
                                                =0

CONJECTURE
Global Regularity for all smooth initial data

STRATEGY
• Reduce to small energy initial data                Is the initial value problem
• Prove global regularity for all smooth data         well-posed in the H1 norm
with small energy.
                 RELATED OPEN PROBLEMS
 4. Small energy implies
                                                       Cosmic Censorship
 regularity-critical case

 : IR 2+1                M                                         Wave maps


     I + IJK ()   J   K = 0

                                                 4+1                 Yang-Mills
 F =  A -   A+ [ A , A ]     in   IR
 D F = 0    F + [ A , F ] = 0



CONJECTURE
Global Regularity for all smooth,           Is the initial value problem
initial data with small energy.              well-posed in the H1 norm
                RELATED OPEN PROBLEMS

5. Strong stability of
                                                      Cosmic Censorship
   the Minkowski space

 (M,g)          R -1/2 R g = 0

 It has to involve, locally, the L2 norm of 3/2       L2 is the only norm
 derivatives of g and 1/2 derivatives of k.           preserved by evolution.


CONJECTURE
 There exists a scale invariant smallness condition such that all
 developments, whose initial data sets              verify it, have
 complete maximal future developments.


 Leads to the issue of developments of initial data sets with low regularity.
               RELATED OPEN PROBLEMS

6. Finite L2 - Curvature Conjecture                         Cosmic Censorship


(M,g)        R -1/2 R g = 0

CONJECTURE
The Bruhat-Geroch result can be extended to initial data sets
(, g, k) with R(g)  L2 and k  L2 .

Recent progress by Chemin-Bahouri,
Tataru, Klainerman-Rodnianski for
quasilinear wave equations. Classical
result requires




Strong connections with problems 5,7 and 8.
               RELATED OPEN PROBLEMS

7. Critical well-posedness for                           Cosmic Censorship
     Wave Maps and Yang-Mills

CONJECTURE                                        WELL POSED
Well posed for                                       • Hs(loc)-initial data 
                                                     local in time, unique
     Hs(loc)- data for any s > sc.
                                                     Hs(loc)-solutions.
                                                     Continuous dependence
Weakly globally well posed for                       on the data:
   s = sc and small initial data                     • strong analytically
                                                     • weak non-analytically

Wave maps in Rn+1          sc =n/2               CRITICAL EXPONENT s = sc
                                                 Hs is invariant under the non-
Yang-Mills in Rn+1        sc =(n-2)/2
                                                 linear scaling of the equations.


There has been a lot of progress in treating the case   s > sc
            RELATED OPEN PROBLEMS

8. Optimal well posedness for                     Problems 1 and 5
other nonlinear wave equations

     •Elasticity
                                                  Quasilinear systems
     •Irrotational compressible fluids
                                                  of wave equations
     •Relativistic strings and membranes
     •Skyrme - Fadeev models



CONJECTURE
Well posed for Hs(loc)- data for any s > sc.

Weakly globally well posed for s = sc and small initial data

				
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