Regularity Theory for Partial Differential Equations with Super-critical Nonlinearities

作者：Vladimir Sverak，明尼苏达大学双城分校，DMS-0200326*Sverak，Vladimir Sverak，明尼苏达大学双城分校的Vladimir Sverak，0200326*复Ginzburg-Landau方程和Landau-Lifschitz方程。这些方程的正则性理论主要由两个因素控制：线性化算子的光滑性和非线性部分产生奇点的倾向。在上述所有情况下，都有一个特定的临界维度(在许多情况下，在这个临界维以下)方程的自然能量估计和线性化算子的光滑化性质足以保证解的正则性。对于大多数临界尺寸以上的方程来说，都是两个。在临界维度以上，情况会更加复杂，因为每个方程的细节都会发挥更大的作用，而且结果也不是很多。这项研究将集中在这一相对未绘制的地区的问题上。偏微分方程解的正则性理论的研究通常受到以下因素的推动：(1)良好的正则性理论可以作为检验偏微分方程解的正则性是否适合于模拟给定现象的重要检验。(2)关于给定偏微分方程解的正则性的信息越多，我们就越有机会在计算机上设计出计算其解的良好的数值方法。我们可以准确地计算其解的方程或多或少就是那些我们有良好的正则性理论的方程，这并不是巧合。(证实“没有什么比一个好的理论更实用了”。)粗略地说，我们对解的了解越多，就越容易避免数值模拟可能陷入的许多陷阱。这项建议中提出的工作将解决具有强非线性的重要方程的正则性理论中的基本开放问题。这些方程中的许多(如纳维斯托克斯方程或朗道-利夫希茨方程)都是相当实用的interest.***************************************

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.***************************************