Currently I am working on a transport equation and have been able to prove the existence and uniqueness of a weak measurable solution to said equation.
I am now working in trying to jot down (with proof) some regularity results. Other papers have stated (without proof) that given a weak solution to the equation
$$\partial_t f(t,\mathbf{x}) + div_{\mathbf{x}}(a(t,\mathbf{x})f(t,\mathbf{x})) = 0$$ $$f(0,\mathbf{x}) = f_0(\mathbf{x})$$ $$t \geq 0, \mathbf{x} \in \mathbb{R}^d$$
with initial datum $f_0$ Lipschitz and $a(t,\mathbf{x})$ (vector valued function) bounded and Lipschitz in $\mathbf{x}$ then the solution $f(t,\mathbf{x})$ is Lipschitz in $\mathbf{x}$.
How is this proven? Moreover, if we impose stronger conditions on $a$ and $f_0$ will that also comply with a stronger regularity of the solution $f$?
Since $a(t,x)$ is Lipschitz in $x$, it generates a flow $\varphi(t_1,t_2,x)$ on your domain according to the equation $$ \frac \partial{\partial t} \varphi(t_1,t,x) = a(t,\varphi(t_1,t,x)) , \quad \varphi(t_t,t_1,x) = x.\tag 1$$ Note that $\varphi(t_1,t_3,x) = \varphi(t_2,t_3,\varphi(t_1,t_2,x))$. From this we obtain that $\varphi(t_1,t_2,\cdot)$ is an invertible function with inverse $\varphi(t_2,t_1,\cdot)$. It is not hard to see that these functions are all Lipschitz in $x$.
Now if you are looking for weak solutions, then the adjoint equation is $$ \frac\partial{\partial t} \theta(t,x) = -a(x,t) \cdot \nabla_x \theta(t,x) , \tag 2$$ where $\theta$ is the test function. This equation has a solution: $$ \theta(t_1,\varphi(t_1,t_2,x)) = \theta(t_2,x) .\tag3$$ Since you can put as much regularity on $\theta(0,\cdot)$ as you like, this formula is rather solid. From this, we can see that the solution to your equation is $$ f(t,x) = \det(D_x \varphi(t,0,x)) f_0(\varphi(t,0,x)) .\tag4$$
I think this should be enough to give you a starting point.
I might have some of the fiddly details wrong, but I think this works in principle.