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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >Stochastic PDEs and Lack of Regularity A Surface Growth Equation with Noise: Existence, Uniqueness, and Blow-up
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Stochastic PDEs and Lack of Regularity A Surface Growth Equation with Noise: Existence, Uniqueness, and Blow-up

机译：随机PDE和不规则性带有噪声的表面增长方程：存在，唯一性和爆炸

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We review results on the existence and uniqueness for a surface growth model with or without space-time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open. In the physically relevant dimension d = 2 and with the physically relevant space-time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of reg-ularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up.
机译：我们回顾了具有或不具有时空白噪声的表面生长模型的存在性和唯一性的结果。如果表面是图形，则该模型在能量估计和缩放性质方面与三维Navier-Stokes方程具有惊人的相似性，并且在这两个模型中，全局弱解的唯一性问题仍然存在。在与物理相关的维数d = 2上，并且由与物理相关的时空白噪声驱动方程，对于温和解的直接定点参数失败​​，因为对于随机强迫没有足够的规律性。这种情况是最简单的情况，可以应用Martin Hairer引入的规则结构方法，尽管在这里我们遵循一种明显简单得多的方法来突出关键问题。使用频谱Galerkin方法或任何其他类型的噪声正则化，可以给随机PDE赋予严格的含义，并在该环境中显示局部解的存在和唯一性。而且，几种类型的正则化似乎可以产生所有相同的解决方案。最后，我们简要评论可能出现的爆炸现象，并通过简单的论证表明，许多复数值解实际上在有限时间内爆炸。这表明仅靠能量估算还不足以验证解决方案的全球唯一性。 Li和Sinai的3D-Navier Stokes已经知道了这个方向的结果，该方法处理的是复杂的值解，而Tao则通过构造带有爆炸的Navier-Stokes类型的方程来解决这一问题。

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