Abstract We study the well-posedness of Cauchy problems on the upper half space R + n + 1 associated to higher order systems ∂ t u = ( − 1 ) m + 1 div m A ∇ m u with bounded measurable and uniformly elliptic coefficients. We address initial data lying in L p ( 1 p ∞ ) and BMO ( p = ∞ ) spaces and work with weak solutions. Our main result is the identification of a new well-posedness class, given for p ∈ ( 1 , ∞ ] by distributions satisfying ∇ m u ∈ T m p , 2 , where T m p , 2 is a parabolic version of the tent space of Coifman–Meyer–Stein. In the range p ∈ [ 2 , ∞ ] , this holds without any further constraints on the operator and for p = ∞ it provides a Carleson measure characterization of BMO with non-autonomous operators. We also prove higher order L p well-posedness, previously only known for the case m = 1 . The uniform L p boundedness of propagators of energy solutions plays an important role in the well-posedness theory and we discover that such bounds hold for p close to 2. This is a consequence of local weak solutions being locally Holder continuous with values in spatial L l o c p for some p > 2 , what is also new for the case m > 1 .

中文翻译：

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具有粗糙系数的抛物线柯西问题的帐篷空间适定性

摘要 我们研究了与高阶系统相关的上半空间 R + n + 1 柯西问题的适定性 ∂ tu = ( − 1 ) m + 1 div m A ∇ mu 具有有界可测和一致椭圆系数。我们处理位于 L p ( 1 p ∞ ) 和 BMO ( p = ∞ ) 空间中的初始数据，并使用弱解。我们的主要结果是识别出一个新的适定类，通过满足 ∇ mu ∈ T mp , 2 的分布为 p ∈ ( 1 , ∞ ] 给出，其中 T mp , 2 是 Coifman 帐篷空间的抛物线版本– Meyer–Stein。在 p ∈ [ 2 , ∞ ] 范围内，这对算子没有任何进一步的约束，并且对于 p = ∞ 它提供了具有非自治算子的 BMO 的 Carleson 测度表征。我们还证明了高阶 L p适定性，以前只知道 m = 1 的情况。