Mathematical Sciences: Properties of Solutions of Linear andNonlinear Partial Integrodifferential Equations

这一数学研究关注的是一类 二阶偏微分方程的产生， 粘弹性固体或液体的描述。 的问题 占据基本抛物线和 扩散现象的偏微分方程， 模拟传播现象的双曲方程。 目标是 建立任意的广义解的存在性 在适当假设下的初始数据，并分析其 特性. 具体地说，渐近空间和时间 这些解决方案的性能有待研究。 问题 方程中固有的阻尼机制是否 可以足够强大，以防止某些不连续的 解决方案（对应于材料不稳定性）， 出现，也将被接受。 具体任务将包括建设薄弱 无界域解与剪切反射问题 粘弹性液体在两块板之间的流动。 还有， 具有不可积核的拟线性方程的解 将被寻求。 逼近解的极限 在这种情况下似乎很容易处理，但尚未得到证实。 所得弱解的实际光滑度也将 需要学习。 第三项具体任务涉及详细分析 变系数线性方程组 有 有证据表明，解决方案实际上可能会获得 平滑他们的数据。 部分标准技术 微分方程（最大值原理，变分 特征）不适用，因此新工具必须 在取得进展之前，我们可以期待。

This mathematical research is concerned with a class of partial differential equations of second order arising in the description of viscoelastic solids or liquids. The problems occupy an intermediate position between the basic parabolic partial differential equations for diffusive phenomena and hyperbolic equations modeling propagative phenomena. The goal is to establish the existence of generalized solutions for arbitrary initial data under suitable assumptions and to analyze their properties. Specifically, the asymptotic spacial and temporal behavior of such solutions is to be studied. The question of whether the damping mechanisms that are inherent in the equations can be powerful enough to prevent certain discontinuities of the solutions (corresponding to material instabilities) from appearing, will also be taken up. Specific tasks will include the construction of weak solutions in unbounded domains and reflecting problems of shear flows of viscoelastic liquids between two plates. Also, solutions of quasilinear equations with nonintegrable kernels will be sought. A passage to the limit of approximate solutions appears tractable in this setting but has yet to be confirmed. The actual smoothness of the resulting weak solution also would require study. A third specific task concerns detailed analysis of linear equations with variable coefficients. There is evidence suggesting that solutions may actually gain in smoothness over their data. Standard techniques from partial differential equations (maximum principle, variational characterizations) do not apply, so that new tools will have to be developed before progress can be expected.