Mathematical Sciences: Free Boundary Problems, Fully Nonlinear Equations, Nonlinear Parabolic Equations and the Navier-Stokes Equations

该项目将涵盖非线性偏微分方程式的问题。关于自由边界问题，正在发展新的方法来研究自由边界的规律性。将沿着De Giorgi和Krylov关于具有有界可测系数的线性方程的相应不等式的路线，为自由边界建立“Harnack”型不等式。计划研究两阶段Stefan问题和极小曲面问题。关于完全非线性方程的工作也将继续进行。理论中的一个主要约束是非线性函数的凹性，因为它依赖于未知函数的二阶导数。计划是调查系数在哪里不一定是连续的。我们将努力得到最优正则性和唯一性定理。研究非线性抛物方程组解的正则性。此外，还将对三维空间中的纳维斯托克斯方程进行研究。早期与Kohn和Nirenberg的工作表明，已知的解应该在弱勒贝格空间中。这将与可能的奇异解的构造一起探索。

The project will cover problems in nonlinear partial differential equations. On free boundary problems, new methods are being developed to study the regularity of free boundaries. Work will be done in establishing "Harnack"-type inequalities for free boundaries along the lines of the corresponding inequalities for linear equations with bounded measurable coefficients by De Giorgi and Krylov. Plans are to study two- phase Stefan problems and minimal surface problems as well. Work will also continue on fully nonlinear equations. A main constraint in the theory is the concavity of the nonlinear function in its dependence on the second derivatives of the unknown function. Plans are to investigate where the coefficients are not necessarily continuous. Efforts will be made to get optimal regularity and uniqueness theorems. The regularity of solutions of nonlinear systems of parabolic equations will be investigated. In addition, work will be done on Navier Stokes equations in three space dimensions. The earlier work with Kohn and Nirenberg suggests that the known solution should be in weak Lebesgue Space. This will be explored along with the possible construction of singular solutions.