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

这项数学研究涉及描述粘弹性固体或液体时产生的一类二阶偏微分方程组。这些问题介于描述扩散现象的基本抛物型偏微分方程组和模拟传播现象的双曲型方程之间。目标是在适当的假设下建立任意初始数据的广义解的存在性，并分析它们的性质。具体地说，这类解的渐近空间和时间行为有待研究。方程中固有的阻尼机制是否足够强大，以防止出现某些解的不连续性(对应于材料不稳定性)，这一问题也将被讨论。具体任务将包括构造无界域中的弱解，以及反映粘弹性液体在两个平板之间的剪切流动问题。此外，还将寻求具有不可积核的拟线性方程的解。在这种情况下，达到近似解的极限似乎是容易的，但尚未得到证实。由此产生的弱解的实际光滑性也需要研究。第三项具体任务涉及对变系数线性方程的详细分析。有证据表明，解决方案实际上可能会提高数据的平稳性。偏微分方程组的标准技术(最大值原理、变分刻画)不适用，因此在取得进展之前必须开发新的工具。

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.