Local and global results for gauge field equations
speaker: Jacob Sterbenz (Princeton University)abstract: SUMMARY OF TALKS GIVEN AT THE SEMESTER FOR PHASE SPACE ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS AT THE RESEARCH CENTER ENNIO DE GIORGI JACOB STERBENZ 1. General Overview The title of my sequence of talks was "Recent progress on hyperbolic gauge field equations". The goal was to give a short introduction with some details to recent work of myself and Matei Machedon on the local and global regularity properties of certain field equations which arise in mathematical physics, particularly those of Yang-Mills type. The focus of these works is the so called "low regularity" approach to dispersive PDE, which is an attempt to understand the long term behavior of equations by proving local well posedness of the Cauchy problem in scale invariant Sobolev (or Besov) spaces. In this way, one is led to discover the regularity properties of the equations in question in spaces which are much rougher than can be treated by the "classica" methods of energy estimates and Sobolev embeddings. The main tools used in this line of investigation are special microlocal decompositions which take into account the specific nature of the symbol of the linear part of the equation being considered, and certain linear and multilinear space-time estimates for the linear part known as Strichartz estimates. 2. Summary of First Talk In the first talk, the main emphasis was on introducing the equations in a geometrically invariant form which emphasizes their tensorial character. The lecture started with a short statement of the basic geometric structure which underlies the Yang-Mills equations and the definition of a Yang-Mills field as the stationary point of an appropriate Lagrangian. I then discussed the specific form of the Euler-Lagrange equations in both a first and second order formulation for the curvature, and then in second order form for the gauge potentials. This led to several equivalent formulations of the Yang-Mills equations. The Cauchy problem was then introduced and there was a short discussion of the constraint equations on the initial data. Finally, I gave a list of some of the most important gauge conditions for the potential formulation of the equation. The next topic in this lecture was the Maxwell-Klein-Gordon (MKG) equations. These serve as a model for much of the analysis that one does on the Yang-Mills system. Again the equations were introduced in geometric form through a Lagrangian, and several equivalent formulations of the equations were given. After the equations had been introduced, the next topic of discussion was the two guiding principles of low regularity analysis: the conservation and scaling properties 1 2 JACOB STERBENZ of the equations. The energy-momentum tensors for the equations were introduced and as a consequence the (main) constants of motion were written down. The scale transformation for the equations was then demonstrated and the scale invariant Sobolev spaces were listed for the various field quantities in various dimensions. The next order of business was the introduction of several sets of model equations for the Yang-Mills and MKG systems. Finally, the talk was ended with a list of progress made by various authors on the low regularity question for the different sets of model equations and for the true MKG equations. 3. Summary of Second Talk The topic of the second talk was my work with Matei Machedon on the almost optimal well-posedness of the (3 + 1) dimensional MKG system. Here we proved that this system is locally well-posed at the level of H 12+. The scaling is the homogeneous Sobolev space H12. The first topic of discussion here was the failure of the MKG model equations to be well posed (at least with respect to Picard iteration) for regularities less than H34. This was shown through the standard reduction of inductive estimates for the system of equations to bilinear estimates for the scalar linear wave equation. The failure of these bilinear estimates was demonstrated through the use of transverse wave packets. I then gave a short discussion of why these wave packets receive extra penalization in the true equations due to a somewhat complicated interaction between the "elliptic" and "hyperbolic" variables in the full MKG equations. All of this was done through calculations of the symbols of specific multi-linear "null structures". 4. Summary of Third Talk The third and final talk I delivered was on recent work concerning global well posedness for model equations which take into account the main interaction in Yang-Mills and related systems. The work here was done in (6 + 1) and higher dimensions, and relies exclusively on linear Strichartz estimates and micro-local decompositions. Since this work is technically much cleaner than that of the previous lecture, an attempt was made to more or less cover the entire result with some detail. The first order of business was to set up the equations in an integral form, and then postulate the solution of the problem through the construction of an auxiliary function space in which one could prove the so called "division estimate". The division estimate was then broken down into a series of pieces through a Littlewood-Paley type decomposition of the non-linearity. It was first shown how the High×High frequency interaction could be handled directly through the L2(L4) Strichartz estimate. The remainder of the talk then focused on the Low × High frequency interaction which represents the technical core of the result. The main tool used to understand this case was presented, which is the microlocal decomposition of a frequency localized product into angular sectors which take place in an essentially diagonal sum. This is very closely related to the so called "second dyadic decomposition" as presented in the talks of Hart Smith. It was then shown how these finer dyadic decompositions can be used in conjunction with the Bernstein 3 inequality from harmonic analysis and the Strichartz estimates to achieve the desired result. Finally, a short discussion was made of special square function norms which are needed to handle certain frequency interactions which cannot be treated directly through the use of Strichartz estimates. Department of Mathematics, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA E-mail address: sterbenz@math.princeton.edu
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