Regularity Theory for Partial Differential Equations with Super-critical Nonlinearities

PI: Vladimir Sverak，明尼苏达大学-双城DMS-0200326***************提出的“具有超临界非线性的偏微分方程的正则性理论”的建议摘要：在该建议中，我们建议研究以下偏微分方程的正则性理论中的开放问题：2.变分学中由多重积分的欧拉-拉格朗日方程引起的强椭圆系统；三维Navier-Stokes方程及相关模型方程；说谎。复金兹堡-朗道方程和朗道-利夫希茨方程。这些方程的正则性理论主要由两个因素决定：线性化算子的平滑性和非线性部分产生奇点的倾向。在上述所有情况下，都有一个临界维，在这个维以下（在许多情况下），方程的自然能量估计和线性化算子的平滑性质足以保证解的正则性。对于上述大多数方程，临界维数是2。在临界维度以上，情况就更加复杂了，因为每个方程的细节都要发挥更大的作用，而已知的结果却不多。研究将集中在这一相对未绘制地图的地区。研究偏微分方程的正则性理论通常受到以下因素的推动：(1)一个好的正则性理论可以作为检验偏微分方程是否适合模拟给定现象的重要手段。(2)对于给定偏微分方程解的规律性，我们掌握的信息越多，我们就越有可能设计出一种好的数值方法来在计算机上计算其解。我们能精确计算出解的方程或多或少与我们有良好规律性理论的方程完全相同，这并非巧合。（证实了“没有什么比一个好的理论更实用”。）粗略地说，我们对解了解得越多，就越容易避免数值模拟可能陷入的一些陷阱。本提案所建议的工作将解决具有强非线性的重要类方程的正则性理论中的基本开放问题。许多这样的方程（如纳维尔-斯托克斯方程，或朗道-利夫希茨方程）具有相当大的实际意义。***************************************

PI: Vladimir Sverak, University of Minnesota - Twin Cities DMS-0200326***************Proposal "Regularity Theory for Partial Differential Equations withSuper-critical Nonlinearities" by V. SverakAbstract: In this proposal we suggest to study open problems in regularitytheory of the following partial differential equations:1. strongly elliptic systems arising as Euler-Lagrange equations for multiple integrals in the Calculus of Variations;2. three-dimensional Navier-Stokes equations and some related model equations; and3. Complex Ginzburg-Landau Equations and Landau-Lifschitz equations.The regularity theory for these equations is mainly governed by twofactors: the smoothing properties of the linearized operator,and the tendency of the non-linear part to produce singularities.In all the cases above there is a certain critical dimensionbelow which (and, in many cases, at which) the natural energyestimates for the equations together with smoothing properties of the linearized operatorare sufficient to guarantee regularity of solutions. For mostof the equations above the critical dimension is two. The situationis more complicated above the critical dimension, when the specificsof each equation come much more into play, and not many resultsare known. The research will concentrate on problems in thisrelatively unmapped area. The study of regularity theory for Partial Differential Equations(PDE) is usually motivated by the following factors:(1) A good regularity theory can serve as an important check that a PDE is appropriate for modeling a given phenomena.(2) The more information we have about regularity of solutions of a given PDE, the better chance we have to design a good numerical method for calculating its solutions on a computer. It is no coincidence that the equations for which we can calculate solutions accurately are more or less exactly those for which we have a good regularity theory. (Confirming that "There is nothing quite so practical as a good theory".) Roughly speaking, the more we know about solutions, the easier it is the avoid some of the many pitfalls into which numerical simulations can fall.The work suggested in this proposal will address basic open problemsin regularity theory for important classes of equations with strongnon-linearities. Many of these equations (such as theNavier-Stokes equations, or the Landau-Lifschitz equations) are ofconsiderable practical interest.***************************************