great problems in nonlinear evolution equations on the analysis of geometric evolution equations...

Great Problems in Great Problems in Nonlinear Evolution Nonlinear Evolution

EquationsEquationsOn the Analysis of Geometric On the Analysis of Geometric

Evolution EquationsEvolution Equations

Sergiu Klainerman

Los Angeles, August 2000

GOOD PROBLEMS ACCORDING TO HILBERTGOOD PROBLEMS ACCORDING TO HILBERTGOOD PROBLEMS ACCORDING TO HILBERTGOOD PROBLEMS ACCORDING TO HILBERT

1. Clear and easy to comprehend

“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 generalizations

Should be related to meaningful simpler problems

If we do not succeed in solving a mathematical problem , the reason is often do to our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems’’

``In dealing with mathematical problems, specialization plays, I believe, a still more important part then generalization. Perhaps in most cases where we seek in vain the 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.’’

2. Difficult yet not completely inaccessible

3. Should provide a strategic height towards a broader goal

TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS

1. PDE AS A UNIFIED SUBJECT1. PDE AS A UNIFIED SUBJECT1. PDE AS A UNIFIED SUBJECT1. PDE AS A UNIFIED SUBJECT

2. REGULARITY OR BREAK-DOWN2. REGULARITY OR BREAK-DOWN 2. REGULARITY OR BREAK-DOWN2. REGULARITY OR BREAK-DOWN

3. MAIN GOALS3. MAIN GOALS3. MAIN GOALS3. MAIN GOALS

4. MAIN OPEN PROBLEMS4. MAIN OPEN PROBLEMS4. MAIN OPEN PROBLEMS4. MAIN OPEN PROBLEMS

5. RELATED OPEN PROBLEMS5. RELATED OPEN PROBLEMS5. RELATED OPEN PROBLEMS5. RELATED OPEN PROBLEMS

PDE AS A UNIFIED SUBJECTPDE AS A UNIFIED SUBJECTPDE AS A UNIFIED SUBJECTPDE AS A UNIFIED SUBJECT

How to generate interesting PDEsHow to generate interesting PDEs


Euclidian Space:Euclidian Space:

= 1

2

+ … + n

2

RRnn Minkowski Space:Minkowski Space: RRn+1n+1

= - t2

+ 1

2

+ … + n

2

Simplest differential operators invariant under

the isometry group

Riemannian Riemannian (M,g)(M,g)

g

= g i j i j

g = g

Lorentzian Lorentzian (M,g)(M,g)




Euler-Lagrange equations

Euler-Lagrange equations

GeometricLagrangian

GeometricLagrangian

Variational Variational principleprinciple

SymmetriesSymmetries Conservation LawsConservation Laws

Variational Variational principleprinciple

Effective equations

Effective equations

Symmetry Reductions

Well-defined Limits

Phenomenological Reductions

• Geometric Geometric (Elliptic)• Mathematical PhysicsMathematical Physics (Hyperbolic)

• Geometric Geometric (Elliptic)• Mathematical PhysicsMathematical Physics (Hyperbolic)


4. Others

• Well-defined LimitsWell-defined Limits Newtonian limit Incompressible limit• Symmetry Reduction Symmetry Reduction • PhenomenologicalPhenomenological

• Well-defined LimitsWell-defined Limits Newtonian limit Incompressible limit• Symmetry Reduction Symmetry Reduction • PhenomenologicalPhenomenological

2.Effective Equations

3.Diffusive Equations

Derived from the fundamental equations by taking limits or making specific simplifying assumptions

• GeometricGeometric Heat flows• Mathematical PhysicsMathematical Physics Stochastic

• GeometricGeometric Heat flows• Mathematical PhysicsMathematical Physics Stochastic

1.Fundamental LawsObtained from a simple geometric Lagrangian


4. Other

2. Effective Equations

3. Diffusive Equations

1. Fundamental Laws Geometrical Equations Geometrical Equations Elliptic •Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

•Ginsburg-Landau

• Seiberg-Witten

• Minimal Surfaces

• Einstein metrics

Geometrical Equations Geometrical Equations Elliptic •Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

•Ginsburg-Landau

• Seiberg-Witten



Equations which play a fundamentalrole in Mathematics . Find objects with optimal geometric properties.

Equations which play a fundamentalrole in Mathematics . Find objects with optimal geometric properties.

Obtained from a simple geometric Lagrangian

Physical EquationsPhysical Equations Hyperbolic

Relativistic Field Theories: • Wave and Klein-Gordon

equations• Maxwell• Wave Maps• Yang Mills• Einstein Field equations

Relativistic Continuum

Mechanics • Elasticity• Gas dynamics• Magneto fluid-dynamics

Physical EquationsPhysical Equations Hyperbolic

Relativistic Field Theories: • Wave and Klein-Gordon

equations• Maxwell• Wave Maps• Yang Mills• Einstein Field equations

Relativistic Continuum

Mechanics • Elasticity• Gas dynamics• Magneto fluid-dynamics

PDE AS A UNIFIED SUBJECTPDE AS A UNIFIED SUBJECT PDE AS A UNIFIED SUBJECTPDE AS A UNIFIED SUBJECT

4. Other



1. Fundamental LawsObtained from a simple geometric Lagrangian

Equations which correspond to our major physical theories

Equations which correspond to our major physical theories


• Scalars

• Connections on a Principal Bundle

• Lorentzian or Riemannian metrics

• Mappings between Manifolds

• Composite Equations

OUR MAIN EQUATIONS

Nonlinear Klein-GordonNonlinear Klein-Gordon

Harmonic and Wave MapsElasticity, Hydrodynamics, MHDMinimal Surface Equation

Harmonic and Wave MapsElasticity, Hydrodynamics, MHDMinimal Surface Equation

Einstein equationsEinstein equations

Yang-MillsYang-Mills

• Well-defined LimitsWell-defined Limits Newtonian limit (non-relativistic)

•Schrödinger• Elasticity• Gas dynamics

Incompressible limitEuler equations

• Symmetry Reductions Symmetry Reductions • stationary• spherically symmetric• dimensional reduction

• PhenomenologicalPhenomenological

•Dispersive( KdV,Schrödinger)

• Ginsburg-Landau

• Maxwell-Vlasov

• Well-defined LimitsWell-defined Limits Newtonian limit (non-relativistic)

•Schrödinger• Elasticity• Gas dynamics

Incompressible limitEuler equations

• Symmetry Reductions Symmetry Reductions • stationary• spherically symmetric• dimensional reduction

• PhenomenologicalPhenomenological

•Dispersive( KdV,Schrödinger)

• Ginsburg-Landau

• Maxwell-Vlasov4. Other



1.Fundamental Laws


Derived from the fundamental equations by taking limits or making specific simplifying assumptions


Parabolic

• Geometrical Equations Geometrical Equations • Ricci Flow• Harmonic Map Flow• Gauss Flow• Mean Curvature Flow• Inverse Mean Curvature Flow

• Physical EquationsPhysical EquationsMacroscopic limit• Compressible Fluids (heat conduction)• Navier-Stokes (viscosity)• Electrodynamics (resistivity)

Parabolic

• Geometrical Equations Geometrical Equations • Ricci Flow• Harmonic Map Flow• Gauss Flow• Mean Curvature Flow• Inverse Mean Curvature Flow

• Physical EquationsPhysical EquationsMacroscopic limit• Compressible Fluids (heat conduction)• Navier-Stokes (viscosity)• Electrodynamics (resistivity)

4. Other



1.Fundamental Laws

CONSERVATION LAWS, A-PRIORI BOUNDS

Noether Theorem: Energy, Linear Momentum Angular Momentum, Charge

•Maximum principle • Monotonicity

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

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

Elliptic and diffusive equations possess additional a-priori estimates.


To any symmetry of the Lagrangean there corresponds a Conservation Law.To any symmetry of the Lagrangean there corresponds a Conservation Law.

Are there other stronger a-priori boundsAre there other stronger a-priori bounds

Solutions to our basic nonlinear Solutions to our basic nonlinear equations, corresponding to equations, corresponding to smooth initial conditions, may smooth initial conditions, may form singularities in finite time,form singularities in finite time,

despite the presence of conserved quantities . .

Solutions to our basic nonlinear Solutions to our basic nonlinear equations, corresponding to equations, corresponding to smooth initial conditions, may smooth initial conditions, may form singularities in finite time,form singularities in finite time,

despite the presence of conserved quantities . .

REGULARITY OR BREAK-DOWN REGULARITY OR BREAK-DOWN REGULARITY OR BREAK-DOWN REGULARITY OR BREAK-DOWN

When?Can solution be continued past the singularities?

Why?What is the character of

the singularities?

Where?

REGULARITY OR BREAK-DOWNREGULARITY OR BREAK-DOWN REGULARITY OR BREAK-DOWNREGULARITY OR BREAK-DOWN

The problem of possible break-down of solutions to The problem of possible break-down of solutions to interesting, non-linear, geometric and physical systems interesting, non-linear, geometric and physical systems is:is:

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.

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.

the most conspicuous unifying problem; it affects all PDE

the most conspicuous unifying problem; it affects all PDE

the most basic problem in PDE

the most basic problem in PDE


SUBCRITICALSUBCRITICALE > NE > N

CRITICALCRITICALE = NE = N

SUPERCRITICALSUPERCRITICALE < NE < N




A-PRIORI BOUNDS(ENERGY)

(E=strength of the bound )

A-PRIORI BOUNDS(ENERGY)

(E=strength of the bound )

SCALING (N=strength of nonlinearity)

SCALING (N=strength of nonlinearity)







Expect global regularity for all data.Expect global regularity for all data.Expect global regularity for all data.Expect global regularity for all data.

Expect, in most cases, global regularity Expect, in most cases, global regularity for all data.for all data.

Expect, in most cases, global regularity Expect, in most cases, global regularity for all data.for all data.

Expect global regularity for “small'' Expect global regularity for “small'' data. data. Expect large data breakdown.Expect large data breakdown.


GENERALGENERALEXPECTATIONSEXPECTATIONS








SUPERCRITICALSUPERCRITICALE < NE < N Expect global regularity for “small'' Expect global regularity for “small''

data. data. Expect large data breakdown.Expect large data breakdown.


GENERALGENERALEXPECTATIONSEXPECTATIONS

What is the character of the What is the character of the breakdown?breakdown?

Can solutions beCan solutions be extended pastextended past

the singularities?the singularities?

MAIN GOALSMAIN GOALSMAIN GOALSMAIN 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.


• 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 solutions, corresponding to all reasonable initial conditions.

• Determine the main asymptotic features of the general solutions.

• 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 solutions, corresponding to all reasonable initial conditions.

• Determine the main asymptotic features of the general solutions.

CAUSALITYCAUSALITY

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


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

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

Should we continue to trust and study them, nevertheless, for pure mathematical reasons?

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

Should we abandon them in favor of the original equations or a better approximation?

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


CALCULUS OF VARIATIONSCALCULUS OF VARIATIONSCALCULUS OF VARIATIONSCALCULUS OF VARIATIONS

Geometrical Equation Geometrical Equation Elliptic •Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

• Saiberg-Witten



Geometrical Equation Geometrical Equation Elliptic •Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

• Saiberg-Witten



EVOLUTION OF EQUATIONSEVOLUTION OF EQUATIONS EVOLUTION OF EQUATIONSEVOLUTION OF EQUATIONS

Geometric Flows Geometric Flows Parabolic

• Ricci •Harmonic Map • Gauss • Mean Curvature • Inverse Mean Curvature



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






Penrose inequality using the inverse mean curvature flow.

Penrose inequality using the inverse mean curvature flow.

To be able to handle its solutions past possible singularities. To find a useful concept of generalized solutions.

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

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

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

EVOLUTION OF EQUATIONSEVOLUTION OF EQUATIONS EVOLUTION OF EQUATIONSEVOLUTION OF EQUATIONS

MAIN OPEN PROBLEMSMAIN OPEN PROBLEMSMAIN OPEN PROBLEMSMAIN 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


EINSTEIN VACUUM EQUATIONS EINSTEIN VACUUM EQUATIONS

(M,g)(M,g) RR -1/2 R g -1/2 R g =0=0

EINSTEIN VACUUM EQUATIONS EINSTEIN VACUUM EQUATIONS

(M,g)(M,g) RR -1/2 R g -1/2 R g =0=0

r→∞

(Σ,g,k)Rg−|k|

2 +(trgk)2 =0

divgk−∇ trgk=0

Initial Data Sets Asymptotic Flatness


Cauchy Development

gij =1+2Mr( )δij +0(r−32)

kij =0(r−52),

Existence and Uniqueness(BRUHAT-GEROCH)

Singularity Theorem (PENROSE)

Global Stability of Minkowski (CHRISTODOULOU-KLAINERMAN)

Any (, g, k) has a unique, future, Maximal Cauchy Development (MCD). It may not be geodesically complete.

Any (, g, k) has a unique, future, Maximal Cauchy Development (MCD). It may not be geodesically complete.

The future MCD of an initial data set

(, g, k ) which admits a trapped surface is geodesically incomplete.

The future MCD of an initial data set

(, g, k ) which admits a trapped surface is geodesically incomplete.

The MCD of an AF initial data set

(, g, k) which verifies a global smallness assumption is geodesically complete. Space-time becomes flat in all directions.

The MCD of an AF initial data set

(, g, k) which verifies a global smallness assumption is geodesically complete. Space-time becomes flat in all directions.



Known ResultsKnown Results

Strong 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

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

Weak Cosmic Censorship

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

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.



11 22 33 44

Spherically Symmetric-Scalar Field Model

(D. Christodoulou)


• Formation of trapped surfaces

• Sharp smallness assumption (implies complete regular solutions). Scale invariant BV space

• Examples of solutions with naked singularities

• Rigorous proof of the weak and strong Cosmic Censorship

• Formation of trapped surfaces

• Sharp smallness assumption (implies complete regular solutions). 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

tt u+u u+u u= - u= -p Rp RRR3 3

div u= 0 div u= 0

tt u+u u+u u= - u= -p Rp RRR3 3

div u= 0 div u= 0

Continuation Theorem(BEALS-KATO-MAJDA)


2 Break-down for 3D Euler Equations

Initial Data (regular) u(0, x)=u0 ( x )


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

The solution can be smoothly continued as long as the vorticity remains uniformly bounded; in fact as long as

Local in time existenceLocal in time existence

||ω(t)||L∞dt 0

T

∫<∞

Vorticity =u


2 Break-down for 3D Euler Equations

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.

tt u+u u+u u= - u= -p p div u= 0 div u= 0

tt u+u u+u u= - u= -p p div u= 0 div u= 0

CONJECTURE

Weak FormWeak FormThere exists:

• a regular data u0,

• a time T* = T* (u0 )> 0 • a smooth uC∞( [0, T* ) R3 )

||(t)||L as t T* .

Weak FormWeak FormThere exists:

• a regular data u0,

• a time T* = T* (u0 )> 0 • a smooth uC∞( [0, T* ) R3 )

||(t)||L as t T* .

Strong FormStrong FormMost regular data lead to such behavior. More precisely the set of initial data which lead to finite time break-down is dense in the set of all regular data with respect to a reasonable topology.

Strong FormStrong FormMost regular data lead to such behavior. More precisely the set of initial data which lead to finite time break-down is dense in the set of all regular data with respect to a reasonable topology.

Most unstable equation.

tt u+u u+u u- u- u = -u = -p p RRRR3 3

div u= 0 div u= 0

tt u+u u+u u- u- u = -u = -p p RRRR3 3

div u= 0 div u= 0

Continuation Theorem(SERRIN)


Initial Data (regular) u(0,x)=u0 (x)




The solution can be smoothly continued as long as the velocity u remains uniformly bounded; in fact as long as

The solution can be smoothly continued as long as the velocity u remains uniformly bounded; in fact as long as

Local in time existence

3 Global Regularity for Navier-Stokes||u(t)||L∞

2dt 0

T

∫<∞


It is however entirely possible that singular solutions exist but are unstable and therefore difficult to construct analytically and impossible to detect numerically.

CONJECTURE

3 Global Regularity for Navier-Stokes

tt u+u u+u u- u- u = -u = -pp div u= 0 div u= 0

tt u+u u+u u- u- u = -u = -pp div u= 0 div u= 0

NOTE OF CAUTION

Break-down requires infinite velocities--unphysical:• incompatible with relativity• thin regions of infinite velocities are incompatible with the assumption of small mean free path required in the macroscopic derivation of the equations.

The solutions corresponding to generic, regular initial data can be continued for all t≥0.

The solutions corresponding to generic, regular initial data can be continued for all t≥0.

The solutions corresponding to all regular initial data can be smoothly continued for all t≥0.

The solutions corresponding to all regular initial data can be smoothly continued for all t≥0.


Initial Data at t=0 =f, t = g

Known FactsKnown Facts


- V’(- V’() = 0, R) = 0, RRR33

V = V = p+1p+1

- V’(- V’() = 0, R) = 0, RRR33

V = V = p+1p+1

Subcritical p < 5

Critical p = 5

Supercritical p > 5

Subcritical p < 5

Critical p = 5

Supercritical p > 5

Global regularity for all data

||φ(t)||L∞p−3

dt 0

T

∫<∞


The solution can be smoothly continuedas long as



- V’(- V’() = 0, R) = 0, RRR33

V = V = p+1p+1

- V’(- V’() = 0, R) = 0, RRR33

V = V = p+1p+1

Subcritical p < 5

Critical p = 5

Supercritical p > 5

Subcritical p < 5

Critical p = 5

Supercritical p > 5

There exist unstable solutions which break down in finite time. Global Regularity for all generic data.

CONJECTURE

Numerical results suggest global regularity for all data.Numerical results suggest global regularity for all data.

5 Global, Singular, Solutions for the 3-D Systems of Conservation Laws

u = (uu = (u11, u, u2 2 ,…, u,…, uNN) ; F) ; F00, F, F11, F, F2 2 ,, FF3 3 : R: RN N R RNN

u = u(t , xu = u(t , x11, x, x22, x, x3 3 )) t t FF00(u)+∑(u)+∑33

i =1i =1 i i FFii(u )= 0(u )= 0

u = (uu = (u11, u, u2 2 ,…, u,…, uNN) ; F) ; F00, F, F11, F, F2 2 ,, FF3 3 : R: RN N R RNN

u = u(t , xu = u(t , x11, x, x22, x, x3 3 )) t t FF00(u)+∑(u)+∑33

i =1i =1 i i FFii(u )= 0(u )= 0


Compressible Euler Equations-ideal gases

Nonlinear Elasticity-hyperelastic materials




There exist arbitrarily small perturbations of the trivial data setwhich break-down in finite time

Local in time existanceLocal in time existance

SingularitiesSingularities(JOHN, SIDERIS)

1-D Global Existence 1-D Global Existence and Uniquenessand Uniqueness(GLIMM, BRESSAN-LIU-YANG)

Global existence and uniqueness forall initial data with small bounded variation.


Find an appropriate concept of generalized solution, compatible with shock waves and other possible singularities, for which we can prove global existence and uniqueness of the initial value problem. For generic data ?



NOTE OF CAUTION

A full treatment of the Compressible Euler equations must include the limiting case of the incompressible equations. This requires not only to settle the break-down conjecture 2 but also a way of continuing the solutions past singularities.

Need to work on vastly simplified modelproblems.

CONCLUSIONS


I. All five problems seem inaccessible at the present time

III. Need to concentrate on simplified model problems

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• They all seem to require the development of generic methods which allow the presence of exceptional sets of data. •Problems 1,4,5 require the development of a powerful hyperbolic theory comparable with the progress made last century in elliptic theory.

The development of such methods may be viewed as

one of the great challengesfor the next century.

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 !

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 PROBLEMSRELATED OPEN PROBLEMSRELATED OPEN PROBLEMSRELATED OPEN PROBLEMS

III. Need to concentrate on simplified model problems:

1. Stability of Kerr1. Stability of Kerr

2. Global Regularity of Space-times with U(1) symmetry2. Global Regularity of Space-times with U(1) symmetry

3. Global regularity of the Wave Maps from R2+1 to H23. Global regularity of the Wave Maps from R2+1 to H2

5. Strong stability of the Minkowski space5. Strong stability of the Minkowski space

6 . Finite L2 - Curvature Conjecture6 . Finite L2 - Curvature Conjecture

7. Critical well-posedness for semi-linear equations 7. Critical well-posedness for semi-linear equations

4. Small energy implies regularity - Critical case4. Small energy implies regularity - Critical case


III. Need to concentrate on simplified model problems:

8. The problem of optimal well- posedness for nonlinear wave and hyperbolic equations8. The problem of optimal well- posedness for nonlinear wave and hyperbolic equations

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

12. Regularity or Break-down for quasi-geostrophic flow12. Regularity or Break-down for quasi-geostrophic flow

9. Global Regularity for the Maxwell-Vlasov equations9. Global Regularity for the Maxwell-Vlasov equations

10. Global Regularity or break-down for the supercritical wave equation with spherical symmetry10. Global Regularity or break-down for the supercritical wave equation with spherical symmetry

1. Stability of Kerr

(M,g)(M,g) RR -1/2 R g -1/2 R g = 0= 0 (M,g)(M,g) RR -1/2 R g -1/2 R g = 0= 0

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 ?

Cosmic Censorship


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

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

CONJECTURE

Compatible with weak

cosmic censorship

(M,g, (M,g, ))

RR == =0=0

(M,g, (M,g, ))

RR == =0=0

2+1 Einstein equations coupled with a wave map with target the hyperbolic space H2.

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

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

Critical !

Cosmic Censorship


polarized U(1)

CONJECTURE

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

Cosmic Censorship


: : IRIR 2+1 2+1 IHIH22

II + + II

JKJK ( () ) JJ KK = = 00

: : IRIR 2+1 2+1 IHIH22

II + + II

JKJK ( () ) JJ KK = = 00

(0)=f, t (0) = g

Global Regularity for all smooth initial data

CONJECTURE

• Reduce to small energy initial data• Prove global regularity for all smooth data with small energy.

STRATEGY

Is the initial value problem well-posed in the H1 norm

4. Small energy implies regularity-critical case

Cosmic Censorship


: : IRIR 2+1 2+1 MM

II + + II

JKJK ( () ) JJ KK = = 00

: : IRIR 2+1 2+1 MM

II + + II

JKJK ( () ) JJ KK = = 00

Global Regularity for all smooth, initial data with small energy.

CONJECTURE

Is the initial value problem well-posed in the H1 norm

FF = =

AA - -

AA+ [ A+ [ A , A, A ] in ] in IRIR 4+1 4+1

DD F F = = 0 0 F F

+ [ A+ [ A

, F, F

]] = = 0 0

FF = =

AA - -

AA+ [ A+ [ A , A, A ] in ] in IRIR 4+1 4+1

DD F F = = 0 0 F F

+ [ A+ [ A

, F, F

]] = = 0 0

Wave mapsWave maps

Yang-MillsYang-Mills

(M,g)(M,g) RR -1/2 R g -1/2 R g = 0 = 0

(M,g)(M,g) RR -1/2 R g -1/2 R g = 0 = 0

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

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

5. Strong stability of the Minkowski space

Cosmic Censorship (Σ,g,k)

It has to involve, locally, the L2 norm of 3/2 derivatives of g and 1/2 derivatives of k.

L2 is the only norm preserved by evolution.


CONJECTURE

The Bruhat-Geroch result can be extended to initial data sets

(, g, k) with R(g) L2 and k L2 .

The Bruhat-Geroch result can be extended to initial data sets

(, g, k) with R(g) L2 and k L2 .

6. Finite L2 - Curvature Conjecture Cosmic Censorship


(M,g)(M,g) RR -1/2 R g -1/2 R g = 0= 0 (M,g)(M,g) RR -1/2 R g -1/2 R g = 0= 0

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

∂β (gαβ(φ)∂βφ) =F(φ,∂φ)

φ(0)= f ∈Hs(Rn),

∂tφ(0) =g∈Hs−1(Rn)

Strong connections with problems 5,7 and 8.

s>sc +1, sc =n2

CONJECTURE

Cosmic Censorship


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

Well posed forWell posed for HHss

(loc)(loc)- data for any - data for any s > ss > scc..

Weakly globally well posed forWeakly globally well posed for s = ss = scc and small initial dataand small initial data

Well posed forWell posed for HHss

(loc)(loc)- data for any - data for any s > ss > scc..

Weakly globally well posed forWeakly globally well posed for s = ss = scc and small initial dataand small initial data

WELL POSED

• Hs(loc)-initial data local

in time, unique Hs(loc)-

solutions. Continuous dependence on the data: • strong analytically• weak non-analytically

CRITICAL EXPONENT s = sc

Hs is invariant under the non-linear scaling of the equations.

Wave maps in RWave maps in Rn+1 n+1 sc =n/2 =n/2

Yang-Mills in RYang-Mills in Rn+1n+1 sc =(n-2)/2 =(n-2)/2

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

CONJECTURE

8. Optimal well posedness for other nonlinear wave equations

Quasilinear systemsof wave equations

Problems 1 and 5


•Elasticity•Irrotational compressible fluids

•Relativistic strings and membranes

•Skyrme - Fadeev models

Well posed for HWell posed for Hss(loc)(loc)- data for any - data for any s > ss > scc..

Weakly globally well posed for Weakly globally well posed for s = ss = scc and small initial dataand small initial data

Well posed for HWell posed for Hss(loc)(loc)- data for any - data for any s > ss > scc..

Weakly globally well posed for Weakly globally well posed for s = ss = scc and small initial dataand small initial data

CONJECTURE

great problems in nonlinear evolution equations on the analysis of geometric evolution equations...

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problems simpler

nonlinear evolution

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