AI 中文总结
本文研究DiPerna–Lions流的空间正则性，在端点p=1时证明其满足定量Osgood型估计，方法基于Karamata慢变函数的加权极大算子，结果可应用于输运方程并得到相关正则性与混合尺度下界。
AI 中文摘要
我们研究DiPerna–Lions类向量场\boldsymbol b∈L^1((0,T);W^{1,1}_{\text{loc}}(\boldsymbol R^d))对应的正则拉格朗日流的空间正则性，该向量场满足标准增长性与可压缩性假设。对于L^1_tW^{1,p}_{\text{loc},x}类（p>1）的向量场，已知流\boldsymbol X(t,·)满足定量局部Lipschitz估计，这意味着它在Lusin意义下是Lipschitz连续的。我们证明，在端点p=1时，该估计存在Osgood型对应结果。更准确地说，我们构造了一个由D\boldsymbol b的可积性确定的递增函数G，满足G(0+)=−∞，使得G(|X(t,x)−X(t,y)|) ≤ G(|X(s,x)−X(s,y)|) + ∫_s^t (k(τ,x)+k(τ,y)) dτ，其中k是局部可积函数。由此可得，流\boldsymbol X(t,·)在任意小测度的集合之外是一致连续的，其显式连续模由D\boldsymbol b的可积性性质确定，所得模包括Hölder、对数Lipschitz regime以及弱得多的Osgood模。我们的方法基于与Karamata意义下慢变函数相关的一类新的加权极大算子。我们还提供例子表明，所得估计在多个方面是最优的，且经典的Lipschitz型估计在端点p=1时可能失效。最后，我们将流估计应用于输运方程，得到输运标量的加权对数Sobolev正则性，以及W^{1,1}情形下泛函和几何混合尺度的对应下界。
英文摘要
We study the spatial regularity of regular Lagrangian flows associated with vector fields in the DiPerna--Lions class \(\boldsymbol b\in L^1((0,T);W^{1,1}_{\loc}(\mathbb R^d)), \) under the standard growth and compressibility assumptions. For vector fields in \(L^1_tW^{1,p}_{\loc,x}\), with \(p>1\), the flow \(\boldsymbol X(t,\cdot)\) is known to satisfy a quantitative local Lipschitz estimate, which implies that it is Lipschitz continuous in the Lusin sense. We prove that, at the endpoint \(p=1\), this estimate admits an Osgood-type counterpart. More precisely, we construct an increasing function \(G\), with \(G(0+)=-\infty\), determined by the integrability of \(D\boldsymbol b\), such that \[ G\bigl(|\boldsymbol X(t,x)-\boldsymbol X(t,y)|\bigr) \leq G\bigl(|\boldsymbol X(s,x)-\boldsymbol X(s,y)|\bigr) + \int_s^t \bigl(k(τ,x)+k(τ,y)\bigr)\,\ddτ, \] where \(k\) is locally integrable. As a consequence, the flow \(\boldsymbol X(t,\cdot)\) is uniformly continuous outside a set of arbitrarily small measure, with an explicit modulus of continuity determined by the integrability properties of \(D\boldsymbol b\). The resulting moduli include Hölder and log-Lipschitz regimes, as well as substantially weaker Osgood moduli. Our approach is based on a new family of weighted maximal operators associated with slowly varying functions in the sense of Karamata. We also provide examples showing that the resulting estimates are sharp in several respects and that the classical Lipschitz-type estimate may fail at the endpoint \(p=1\). Finally, we apply the flow estimates to transport equations, obtaining weighted logarithmic Sobolev regularity for transported scalars and corresponding lower bounds on functional and geometric mixing scales in the \(W^{1,1}\) setting.
Comments51 pages
