# Remarks on PDE's

Document Sample

```					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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