We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form \(\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u\) on \([0,T]\times \mathbb {R}^{d}.\) To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces W^k,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u↦F(u), where F is C² and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F(u) = |u|^p with p ≥ 2, but also when Y ≠ 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of L^p-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.

我们研究形式为\（\ partial _ {t} u-A_ {t} uf = {\ dot X_ {t}（x）\ cdot \ nabla + \ dot Y_的粗糙抛物方程解的存在性，唯一性和正则性{ （0，T] \ times \ mathbb {R} ^ {d}。\）上的{t}（x））u \）为此，我们引入了“差分粗略驱动器”的概念在Sobolev空间W ^k，p的基础上建立的，与粗糙路径理论中通常控制路径空间相对应的空间。在这种情况下，我们还定义了自然的几何概念，并显示了它与受控路径的乘积公式之间的关系。在运输噪声的情况下（即，当Y = 0时），我们使用此框架来证明形式的Nemytskii运算的Itô公式（就链规则而言）û ↦ ˚F（Û），其中˚F是Ç ²和在原点消失。我们的方法基于能量估计和Moser迭代参数的一般化，以证明上述抛物线问题的密集类解的有界性。特别是，我们避免使用流变换，而直接在原始方程式的层次上进行工作。我们还显示了F（u）= |的对应链规则。ü | ^p与p ≥2时，而且当ÿ ≠0且p ≥4.如这些结果的应用中，我们证明合适的类的存在唯一性具有乘性噪声的抛物线方程的L ^p解。另一个相关的发展是光滑域上的齐次Dirichlet边界问题，对于该问题，在适当的系数假设下显示了弱最大原理。