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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 “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 2M 0(r 3 2 ) ) k ,g , ( gij 1 r ij 2 2 Rg | k | (trg k) 0 5 r kij 0(r 2 ), divgk trgk 0 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 RR3 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 td || ) t( || 0 L T 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 uC∞( [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 RR3 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 td L2|| ) t(u || 0 T 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, RR3 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 0 || ) t ( || L td 3 p T MAIN OPEN PROBLEMS 4 Global Regularity for other Supercritical Equations - V’() = 0, RR3 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 ) k ,g , ( 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, (g ( ) ) F( , ) Tataru, Klainerman-Rodnianski for (0) f H s (Rn ), quasilinear wave equations. Classical result requires s sc 1, sc n 2 t (0) g H s1 (R n ) 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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Department of Mathematics, Richard Haberman, Student Body, University of Connecticut, Computational Challenges, Universität zu Köln, Worcester Polytechnic Institute, wave equations, math majors, Applied Math

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posted: | 5/13/2010 |

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