For transport equation with constant velocity $v$,
$$u_t+v\cdot \nabla u=0, u(x,0)=f(x)$$
By method of characteristics, we have $u(x,t)=f(x-vt)$. Thus
$$||u(x,t)||_{L^\infty}=||f||_{L^\infty}$$
i.e. the transport eqaution preserves the $L^\infty$ norm.
My question is, when $v=v(x,t)$ becomes non-constant, does the norm-preserving property still hold in general? Do we have to pose some regularity conditions for $v(x,t)$, like Lipschitz continuous as in Cauchy-Lipschitz theorem?
Best Answer
Since you did not specify a domain I will work on $\mathbb{R}$ and assume the initial data $f \in L^{\infty}(\mathbb{R})$. In general the answer is no; if $v$ has low regularity then the equivalence between the characteristics and the transport equation breaks down and uniqueness does not necessarily hold. One can construct an example where the $L^{\infty}$ norm is not preserved.
Suppose $v(t,x)=sgn(x)$ and $f = c \in \mathbb{R} $. Then the function $$u(x,t)= \begin{cases} c & \text{ if } t \leq |x|, \\ c+t-|x| & \text{ if } t \geq |x|. \end{cases}$$ solves $\partial_{t}u+v(t,x)\partial_{x}u=0$. Indeed, for $t \le |x|$ we have $\partial_{t}u=\partial_{x}u=0$ and when $t \ge |x|$ we have $\partial_{t}u=1, ~\partial_{x}u = -\frac{x}{|x|} $ for $x \ne 0$ so $\partial_{t}u + sgn(x)\partial_{x}u = 1 -1 = 0$. Notice additionally that $u$ is continuous and solves the transport equation a.e. It is easy to see that $\|u\|_{L^{\infty}_{t,x}} > c$ if we look at the region $t > |x|$.
One possible condition for your property to hold is to require $v \in C^{1}(\mathbb{R} \times \mathbb{R})$. Then you can use your characteristic method to obtain the solution that you specified. But here is another approach which does not explicitly use characteristics, and hopefully will illustrate why we need some condition on $v$.
We introduce the flow map which for a fixed $x \in \mathbb{R}$ is defined as the solution $X(\cdot, x)$ to the ODE
$$\begin{cases} \displaystyle \frac{dX}{dt} = v(t,X(t,x)), \\ X(0,x)=x . \end{cases}$$ Existence and uniqueness is guaranteed by Cauchy-Lipschitz. Next notice that $$\frac{d}{dt}u(t,X(t,x)) = \partial_{t}u + v\partial_{x}u $$ by the chain rule and so our transport equation is equivalent to $$ \frac{d}{dt}u(t,X(t,x)) = 0.$$ Hence we get $$u(t,X(t,x)) = u(0,X(0,x)) = u(0,x) = f(x).$$ Inverting the flow map (this is where we use $v \in C^{1}$), we get $$u(t,x) = f(X(t,x)) $$ and so $$\|u\|_{L^{\infty}_{t,x}} = \|f\|_{L^{\infty}_{t,x}}. $$ In continuum mechanics, the composition with the flow map is often called a change of variables to Lagrangian coordinates.
I believe one can lower the regularity of $v$ and still use the same argument with the flow map, but I cannot recall the paper where I saw this. Anyway, two famous papers that cover the regularity theory for the transport equation are by DiPerna-Lions, Ambrosio. You might find them interesting. I do not know a lot about this topic so if somebody else knows more modern references feel free to add them in the comments :)