含水平热梯度和热传导的二维可压缩流体模型柯西问题的强解

Strong Solutions to the 2D Cauchy Problem for a Compressible Fluid Model with Horizontal Thermal Gradient and Heat Conduction

摘要: 本文研究了受水平热梯度效应与热传导影响的二维可压缩流体模型的柯西问题。与现有文献中忽略热传导的简化模型不同，热传导项的引入在真空远场条件下引发了新的困难：如何控制温度梯度项的增长。为此，本文通过构造合适的权重函数并建立关于温度及其时间导数的新先验估计，有效克服了方程间的耦合效应。在适当的初值和相容性条件下，我们证明了该模型强解的局部存在性和正则性。这一工作将现有理论推广至更一般的物理框架，也为该模型在相关物理过程中的数值模拟与理论分析提供了坚实的数学基础。

Abstract: This paper investigates the Cauchy problem for a two-dimensional compressible fluid model subject to horizontal thermal gradient effects and heat conduction. In contrast to simplified models in the existing literature that neglect heat conduction, the inclusion of the heat conduction term introduces new challenges under vacuum far-field conditions—specifically, controlling the growth of temperature gradient terms. To address this, we construct suitable weight functions and establish novel a priori estimates for both the temperature and its time derivative, thereby effectively overcoming the strong coupling among the equations. Under appropriate initial conditions and compatibility assumptions, we prove the local existence and regularity of strong solutions to the model. This work extends the current theory to a more general physical framework and provides a rigorous mathematical foundation for numerical simulations and further theoretical analysis of related physical processes.