In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix $A(u)$ are given, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since $A(u)$ is typically neither symmetric nor positive definite. In particular, the normal ellipticity of $A(u)$ for all u and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If A is constant or constant up to nonlinear perturbations, the existence of an entropy structure is equivalent to the normal ellipticity of A. The results are applied to various examples from physics and biology. Finally, the normal ellipticity of the n-species population model of Shigesada, Kawasaki, and Teramoto is proved.

在本说明中，对于具有扩散矩阵的某些类型的交叉扩散系统而言，存在熵结构的充要条件 $\mathrm{一种}（ü）$根据矩阵分解的结果给出。熵结构在此类方程式的分析中很重要，因为$\mathrm{一种}（ü）$通常既不是对称的也不是正定的。特别是椭圆的正常椭圆度$\mathrm{一种}（ü）$对于所有u，Onsager矩阵的对称性表示其正定性，因此具有熵结构。如果A为常数或直到非线性扰动不变，则熵结构的存在等效于A的正常椭圆率。结果被应用于来自物理学和生物学的各种例子。最后，证明了Shigesada，Kawasaki和Teramoto的n种种群模型的正常椭圆性。