具粗糙系数的抛物型方程的柯西问题
On Cauchy Problems for Parabolic Equations with Rough Coefficients
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- School of Mathematical Sciences（复旦大学数学科学学院；黎曼几何与动力系统研究中心及当代应用数学上海市重点实验室）
- LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University
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中文总结 AI 辅助
本文研究扩散系数粗糙的两类抛物型方程柯西问题，在最优分数阶Sobolev空间中建立其唯一可解性，构造近似格式并结合多种分析工具完成估计，证明阈值$s < 1/2$的尖锐性，提供相关数学框架。
中文摘要 AI 辅助
本研究探讨了扩散系数粗糙的非散度型与散度型抛物型方程的柯西问题，这类问题常见于复合介质、金融定价及粘性流体领域。在临界正则性条件下，我们在最优分数阶Sobolev空间中建立了问题的唯一可解性。首个关键技术步骤是构造带有截断和磨光扩散系数$a^\epsilon(t,x)$的有效近似格式，我们严格证明该格式能一致保持高频小性。通过将此近似格式与 paraproduct 分解、Fefferman-Stein 极大不等式、Coifman-Meyer 双线性估计及精细 Sobolev 嵌入相结合，我们完成了一致先验估计并取极限。此外，我们通过构造显式反例证明阈值$s < \frac{1}{2}$是尖锐的。这些理论结果为研究具粗糙系数的抛物型方程柯西问题提供了严格的数学框架。
英文摘要
The presented work investigates the Cauchy problems for parabolic equations in both non-divergence and divergence forms with rough diffusion coefficients, which commonly arise in composite media, financial pricing, and viscous fluids. Under critical regularity settings, we establish the unique solvability in optimal fractional Sobolev spaces. The first key technical step lies in constructing an effective approximation scheme with truncated and mollified diffusion coefficients $a^ε(t,x)$, for which we rigorously prove the uniform preservation of high-frequency smallness. By incorporating this approximation scheme with paraproduct decomposition, Fefferman-Stein maximal inequalities, Coifman-Meyer bilinear estimates, and refined Sobolev embeddings, we close the uniform a priori estimates and then pass to the limit. Furthermore, we prove that the threshold $s < \frac{1}{2}$ is sharp by constructing explicit counterexamples. These theoretical results provide a rigorous mathematical framework in the study of Cauchy problems for parabolic equations with rough coefficients.
