Mathematical Sciences: Nonlinear Partial Differential Equations

这项数学工作的主要重点将集中在三个问题领域：非线性输运方程，变分中的正则性理论，以及运动超曲面的平均曲率几何性质。所有人都以这样或那样的方式与分析中的非线性问题有关。非线性输运可用相对简单的偏微分方程组(如标量守恒定律或多孔介质方程)来模拟。这里的目的是使用粘性解的新概念，并将其作用从由最大范数度量的空间扩展到可积函数的勒贝格空间，以努力理解解。关键是引入满足增生条件的微分算子的弱解，期望得到对非光滑解的解释。这项工作的一个长期目标是分析随机相互作用粒子系统的流体动力学极限。第二类研究涉及变分中向量值问题的部分正则性问题，特别是非线性弹性问题。这里的基本问题是在约束内执行变分，这要求微分的行列式是正的。工作将首先进行更容易处理的问题，涉及流形之间映射的凸积分的最小化。目前正在进行大量的活动，用平均曲率研究运动的几何问题。相关的偏微分方程组给出了水平集(流形的)以与平均曲率成比例的速度正常运动的运动，可能有奇异性的解。这项工作的第一个关注点将是证明水平集是光滑的，除了可能可以忽略的奇异集。

Primary emphasis of this mathematical work will be directed at three problem areas: nonlinear transport equations, regularity theory in the calculus of variations, and geometric properties for motion hypersurfaces by mean curvature. All, in one way or another, are concerned with nonlinear problems in analysis. Nonlinear transport is modeled by relatively simple partial differential equations (e.g. scalar conservation laws or the porous medium equation). The goal here is to use the newer concept of viscosity solution and expand its role from spaces metrized by maximum norms to the Lebesgue space of integrable functions, in an effort to understand solutions. The key is the introduction of weak solutions of differential operators satisfying an accretive condition, with the expectation that an interpretation of non-smooth solutions will result. A long-term goal of this work is that of analyzing the hydrodynamical limit of stochastic interacting particle systems. A second line of research concerns the question of partial regularity for vector-valued problems in the calculus of variations - in particular, for problems of nonlinear elasticity. The basic issue here is that of performing variations within the constraint which requires that the determinant of the differential be positive. Work will proceed first with more tractable questions involving minimizers of convex integrands for mappings between manifolds. Considerable activity is presently under way investigating geometric problems of motion by mean curvature. The associated partial differential equations, which give the motion of level sets (of a manifold) moving normally at a velocity proportional to the mean curvature, may have solutions with singularities. The first concern of this work will be to show that the level sets are smooth except possibly for negligible singular sets.