We prove that the solution map for a family of non-linear transport equations in R n {R}ⁿ with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree − (n − 1) - (n-1) is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.
Marc Magaña (Wed,) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: