We study the Cauchy problem for the equations describing a viscous compressible and heat-conductive fluid in two dimensions. By imposing a weight function to initial density to deal with Sobolev embedding in critical space, and constructing an ad-hoc truncation to control the quadratic nonlinearity appeared in energy equation, we establish the local in time existence of unique strong solution with large initial data. The vacuum state at infinity or the compactly supported density is permitted. Moreover, we provide a different approach and slightly improve the weighted $ L^{p} $ estimates in [19,Theorem B.1].

中文翻译：

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二维完全可压缩纳维-斯托克斯方程柯西问题强解的存在性

我们研究了用于描述二维粘性可压缩导热流体的方程的柯西问题。通过对初始密度施加一个权重函数来处理临界空间中的 Sobolev 嵌入，并构造一个临时截断来控制能量方程中出现的二次非线性，我们建立了具有大初始数据的唯一强解的局部时间存在性。允许在无穷远处的真空状态或紧密支撑的密度。此外，我们提供了一种不同的方法，并略微改进了 [ 中的加权 $ L^{p} $ 估计值19，定理 B.1]。