AI 中文总结
本文研究带散度系数的对称输运方程组，证明满足线性增长条件的局部利普希茨系数下其空间输运算子为反对称，对应柯西问题解唯一，且标量输运方程在DiPerna-Lions条件下结论仍成立。
AI 中文摘要
我们研究具有散度系数的对称输运方程组，该方程组可归为实平方可积向量函数希尔伯特空间中带反对称空间算子的演化方程，根据一般结论，柯西问题总存在广义解。该解的唯一性等价于空间算子的反对称性。我们证明，在满足线性增长条件的局部利普希茨系数下，空间输运算子确为反对称；对于标量输运方程，在更弱的DiPerna-Lions条件下该结论仍成立。
英文摘要
We study a symmetric system of transport equations with solenoidal coefficients. This system reduces to an evolutionary equation with a skew-symmetric spatial operator in the real Hilbert space of square-integrable vector-functions, and by general results we claim that there always exists a generalized solution of the Cauchy problem. Uniqueness of this solution is equivalent to skew-adjointness of the spatial operator. We demonstrate that in the case of locally Lipschitz coefficients satisfying a linear growth condition the spatial transport operator is indeed skew-adjoint. For scalar transport equation this result remains true under the weaker DiPerna-Lions conditions.
