Mathematical Sciences: Nonlinear Partial Differential Equations

这项数学工作的主要重点将是 针对三个问题领域：非线性传输方程， 变分法中的正则性理论，以及几何 运动超曲面的平均曲率性质。 所有人 无论如何，都与非线性问题有关， 分析. 非线性输运由相对简单的 偏微分方程（例如标量守恒定律 或多孔介质方程）。 这里的目标是使用 粘度解决方案的新概念，并扩大其作用， 的Lebesgue空间的最大范数度量的空间 可积函数，以便理解解决方案。 关键是引入微分方程的弱解 满足增生条件的算子， 期望非光滑解的解释 将导致。 这项工作的长期目标是， 分析了随机相互作用的流体动力学极限 粒子系统 第二条研究路线涉及部分 向量值问题的正则性 变化-特别是对于非线性问题， 弹性 这里的基本问题是 约束内的变化，这要求 微分的行列式是正数。 工作将继续进行 首先是更容易处理的问题，涉及最小化 流形间映射的凸被积函数。 目前正在开展大量活动， 平均曲率运动的几何问题。 相关联的 偏微分方程，给出水平运动 （流形的）以与速度成比例的速度正常运动的集合 的平均曲率，可能有奇异的解决方案。 这项工作的第一个关注点将是表明， 集合是光滑的，除了可能是可忽略的奇异集合。

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.