# Conservated quantity and hyperbolic equation

Given the hyperbolic Vlasov equation $$\frac{\partial f }{\partial t} +v\nabla_x f + F(t,x)\nabla_vf =0$$ where $$f=f(t,x,v)$$ and $$(t,x,v)\in \mathbb{ R}\times\mathbb{R}^{n}\times \mathbb{R}^{n}$$. I wonder how can be proved that $$\Vert f(t,x,v)\Vert_{L^p(\mathbb{R}^{2n})} = \Vert f(0,x,v)\Vert_{L^p(\mathbb{R}^{2n})}, \quad p\in [1,\infty]$$ Any hint is welcome. Thank in advance.

• I see you've posted a bunch of PDE questions; I'd suggest including the arXiv tag ap.analysis-of-pdes to them if you do so in the future. Oct 28 '19 at 19:36
• Thanks for that suggestion, I will add that tag for future questions. Oct 28 '19 at 19:42

The "one phrase answer" is "divergence theorem".

Slightly wordier but a bit formally (for ease of typing I write $$dz = dx~dv$$ for the volume on phase space)

$$\partial_t \int f^p dz = \int \partial_t f^p dz$$

Next,

$$0 = \int \nabla_x \cdot (vf^p) dz$$

assuming $$f$$ decays suitably fast at infinity, and similarly

$$0 = \int \nabla_v \cdot (Ff^p) dz$$

Now,

$$\partial_t f^p + \nabla_x \cdot (vf^p) + \nabla_v \cdot (F f^p) = p f^{p-1} \left[ \partial_t + v\cdot \nabla_x + F \cdot \nabla_v \right]f = 0$$

($$\nabla_x$$ trivially acts on $$v$$ and $$\nabla_v$$ trivially acts on $$F(t,x)$$) and the result follows.

To be precise, one has to interpret $$|f|^p = \lim_{\epsilon\to 0} (\sqrt{\epsilon^2 + |f|^2}- \epsilon)^{p}$$, and approximate your $$f\in L^p$$ with $$f \in \mathcal{S}\cap L^p$$ or similar.

For the case $$p = \infty$$ it suffices to notice that $$f$$ is constant on the integral curves of the vector field $$(1, v, F(t,x))$$ on $$\mathbb{R}\times\mathbb{R}^n \times \mathbb{R}^n$$.

Finally, something stronger is true: let $$G:\mathbb{R}\to\mathbb{R}$$ be smooth, then $$\int G\circ f ~dz$$ is invariant in time if $$f$$ solves Vlasov.

• Wow! Thank you. I was really stuck... +1 Oct 28 '19 at 19:42