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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.

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Great Problems in Nonlinear Evolution Equations

On the Analysis of Geometric Evolution Equations

Los Angeles, August 2000

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 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.’’

3. Should provide a strategic height towards a broader goal

1. PDE AS A UNIFIED SUBJECT

2. REGULARITY OR BREAK-DOWN

3. MAIN GOALS

4. MAIN OPEN PROBLEMS

5. RELATED OPEN PROBLEMS

(M,g)

g= g i j ij

g = g   

Lorentzian

(M,g)

PDE AS A UNIFIED SUBJECT

?

• How to generate interesting PDEs

Rn+1

Rn

Minkowski Space:

Euclidian Space:

= - t2+ 12+ … + n2

= 12+ … + n2

Simplest differential

operators invariant under

the isometry group

?

• How to generate interesting PDEs

Symmetries

Conservation Laws

GeometricLagrangian

Euler-Lagrange equations

Variational

principle

Variational

principle

Well-defined Limits

Effective equations

Symmetry Reductions

Phenomenological Reductions

1.Fundamental Laws

• Geometric (Elliptic)

• Mathematical Physics(Hyperbolic)

Obtained from a simple geometric Lagrangian

2.Effective Equations

• Well-defined LimitsNewtonian limit Incompressible limit

• Symmetry Reduction

• Phenomenological

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

3.Diffusive Equations

• GeometricHeat flows

• Mathematical PhysicsStochastic

4.Others

1. Fundamental Laws

• Geometrical Equations Elliptic

• Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

• Ginsburg-Landau

• Seiberg-Witten

• Minimal Surfaces

• Einstein metrics

Obtained from a simple geometric Lagrangian

Equations which play a fundamental

role in Mathematics . Find objects

with optimal geometric properties.

2. Effective Equations

3. Diffusive Equations

4. Other

1. Fundamental Laws

• Physical EquationsHyperbolic

• 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

Obtained from a simple geometric Lagrangian

Equations which correspond to

our major physical theories

2. Effective Equations

3. Diffusive Equations

4. Other

OUR MAIN EQUATIONS

• Scalars

• Connections on a Principal Bundle

• Lorentzian or Riemannian metrics

• Mappings between Manifolds

• Composite Equations

Nonlinear Klein-Gordon

Yang-Mills

Einstein equations

Harmonic and Wave Maps

Elasticity, Hydrodynamics, MHD

Minimal Surface Equation

• Well-defined Limits

• Newtonian limit(non-relativistic)

• Schrödinger

• Elasticity

• Gas dynamics

• Incompressible limitEuler equations

• Symmetry Reductions

• stationary

• spherically symmetric

• dimensional reduction

• Phenomenological

• Dispersive( KdV,Schrödinger)

• Ginsburg-Landau

• Maxwell-Vlasov

• 1.Fundamental Laws

2. Effective Equations

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

3. Diffusive Equations

4. Other

1.Fundamental Laws

• Parabolic

• Geometrical Equations

• Ricci Flow

• Harmonic Map Flow

• Gauss Flow

• Mean Curvature Flow

• Inverse Mean Curvature Flow

• Physical Equations

• Macroscopic limit

• Compressible Fluids (heat

• conduction)

• Navier-Stokes (viscosity)

• Electrodynamics (resistivity)

2. Effective Equations

3. Diffusive Equations

4. Other

CONSERVATION LAWS, A-PRIORI BOUNDS

Noether Theorem:

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

Energy,

Linear Momentum

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 estimates.

• Maximum principle

• Monotonicity

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

• despite the presence of conserved quantities .

What is the

character of

the singularities?

Why?

Where?

When?

Can solution be continued past the singularities?

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

• the most basic problem in PDE

• the most conspicuous unifying 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.

• SUBCRITICAL

• E > N

• CRITICAL

• E = N

• SUPERCRITICAL

• E < N

A-PRIORI BOUNDS(ENERGY)

(E=strength of the bound )

SCALING

(N=strength of nonlinearity)

GENERAL

EXPECTATIONS

• SUBCRITICAL

• E > N

• CRITICAL

• E = N

• SUPERCRITICAL

• E < N

Expect global regularity for all data.

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

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

GENERAL

EXPECTATIONS

• SUBCRITICAL

• E > N

• CRITICAL

• E = N

• SUPERCRITICAL

• E < N

?

What is the character of the breakdown?

Can solutions be

extended past

the singularities?

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

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.

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 solutions, corresponding to all reasonable initial conditions.

• Determine the main asymptotic features of the general solutions.

CAUSALITY

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 and study them, nevertheless, for pure mathematical reasons?

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

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

• Cauchy- Riemann

• Laplace

• Dirac

• Hodge systems

• Harmonic Maps

• Yang- Mills

• Saiberg-Witten

• Minimal Surfaces

• Einstein metrics

• 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.

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

EVOLUTION OF EQUATIONS

• Geometric Flows Parabolic

• Ricci

• Harmonic Map

• Gauss

• Mean Curvature

• Inverse Mean Curvature

Penrose inequality using the inverse mean curvature flow.

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

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

1 Cosmic Censorship in General Relativity

EINSTEIN VACUUM EQUATIONS

(M,g)R -1/2 R g =0

Initial Data Sets

Asymptotic Flatness

Cauchy

Development

1 Cosmic Censorship in General Relativity

Known Results

Existence and Uniqueness

(BRUHAT-GEROCH)

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

Singularity Theorem (PENROSE)

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

Global Stability of Minkowski (CHRISTODOULOU-KLAINERMAN)

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.

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.

1 Cosmic Censorship in General Relativity

1

2

3

4

1 Cosmic Censorship in General Relativity

Known Results

• 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

Spherically Symmetric-Scalar Field Model

(D. Christodoulou)

Results for U(1)U(1) symmetries

and Bianchi type

2 Break-down for 3D Euler Equations

tu+u  u= -p RR3

div u= 0

Initial Data (regular) 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.

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

Continuation Theorem

(BEALS-KATO-MAJDA)

Vorticity =u

2 Break-down for 3D Euler Equations

tu+u  u= -p

div u= 0

Most unstable equation.

CONJECTURE

• Weak Form

• There exists:

• a regular data u0,

• a time T* =T* (u0 )> 0

• a smooth uC∞( [0, T* ) R3 )

• ||(t)||L  as t  T* .

Strong Form

Most 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.

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.

3 Global Regularity for Navier-Stokes

tu+u  u- u = -p RR3

div u= 0

Initial Data (regular) 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.

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

Continuation Theorem

(SERRIN)

3 Global Regularity for Navier-Stokes

tu+u  u- u = -p

div u= 0

CONJECTURE

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

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

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

4 Global Regularity for other Supercritical Equations

•  - V’() = 0, RR3

• V =  p+1

Initial Data at t=0 =f, t  =g

Subcritical p < 5

Critical p = 5

Supercritical p > 5

Known Facts

Global regularity for all data

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

4 Global Regularity for other Supercritical Equations

•  - V’() = 0, RR3

• V =  p+1

Subcritical p < 5

Critical p = 5

Supercritical p > 5

Numerical results suggest

global regularity for all data.

CONJECTURE

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

generic data.

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

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.

There exist arbitrarily small perturbations of the trivial data set

which break-down in finite time

Singularities

(JOHN, SIDERIS)

Global existence and uniqueness for

all initial data with small bounded variation.

1-D Global Existence

and Uniqueness

(GLIMM, BRESSAN-LIU-YANG)

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

Conservation Laws

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 model

• 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

• 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 challenges

for the next century.

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 !

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

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

Cosmic Censorship

1. Stability of Kerr

(M,g)R -1/2 R g = 0

CONJECTURE

Compatible

with weak

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.

?

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 ?

2. Global Regularity of

Space-times with U(1)

symmetry

Cosmic Censorship

(M,g, )

R = 

 =0

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

Critical !

polarized U(1)

CONJECTURE

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

3. Global regularity of

the 2+1 Wave Maps

to hyperbolic space.

Cosmic Censorship

• : IR2+1 IH2

• I+ IJK ()  J K =0

(0)=f, t (0) =g

CONJECTURE

Global Regularity for all smooth initial data

?

STRATEGY

• Reduce to small energy initial data

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

Is the initial value problem

well-posed in the H1 norm

4. Small energy implies regularity-critical case

Cosmic Censorship

• : IR2+1 M

• I+ IJK ()  J K =0

Wave maps

Yang-Mills

• F= A- A+ [ A, A] in IR4+1

• D F= 0    F+ [ A, F]= 0

?

CONJECTURE

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

Is the initial value problem

well-posed in the H1 norm

5. Strong stability of

the Minkowski space

Cosmic Censorship

(M,g)R -1/2 R g = 0

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

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.

Cosmic Censorship

6. Finite L2 - Curvature Conjecture

(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.

7. Critical well-posedness for

Wave Maps and Yang-Mills

Cosmic Censorship

CONJECTURE

• WELL POSED

• Hs(loc)-initial data local in time, unique Hs(loc)-solutions. Continuous dependence on the data:

• strong analytically

• weak non-analytically

• Well posed for

• Hs(loc)- data for any s > sc.

• Weakly globally well posed for

• s = scand small initial data

CRITICAL EXPONENT s = sc

Hsis invariant under the non-linear scaling of the equations.

Wave maps in Rn+1 sc=n/2

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

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

8. Optimal well posedness for other nonlinear wave equations

Problems 1 and 5

• Elasticity

Quasilinear systems

of wave equations

• Irrotational compressible fluids

• Relativistic strings and membranes

• Skyrme - Fadeev models

CONJECTURE

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

Weakly globally well posed for s = scand small initial data