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Let $E$ be a fondamental solution of $\partial_{x_1}\square$. Then you have for $u$ compactly supported $$ u=u\ast \delta=u\ast (\partial_{x_1}\square E)= (\partial_{x_1}\square u)\ast E, $$ so that $ …

A semi-linear PDE reads $ \mathcal Lu=F(u), $ where $\mathcal L$ is a linear operator and $F$ is a function. A quasi-linear PDE with order $m$ reads $ \mathcal L\bigl((\partial_x^\alpha u)_{\vert \alp …

Let me change slightly your notations with the flow $\psi (t,y)$ defined by $$ \dot \psi(t,y)=v(t, \psi(t,y)),\quad \psi(0,y)=y. $$ The solution $u$ is constant along the integral curves of the vector …

Yes, as for any strictly hyperbolic equation you do have uniqueness and even much better, well-posedness: you can control the Sobolev norm of $u(t)$ by the Sobolev norms of the initial data $u(0), \do …

Here is what I believe is a relevant example: consider the vector field $X$ in $\mathbb R^3$, $$ X=a_1(x_2, x_3)\frac{\partial}{\partial x_1}+a_2(x_1, x_3)\frac{\partial}{\partial x_2}+a_3(x_1,x_2)\fr …

More (too long) a comment than an answer. It is somewhat paradoxical that almost 90 years after Serguei Sobolev introduced the notion of weak solutions of PDE, also almost 70 years after Laurent Schwa …

You have then $ \rho^2u_{\rho\rho}+\rho u_{\rho}=\rho^2 a^{-2} u_{tt}. $ Looking for a solution of the form $\phi(t) w(\rho)$, we obtain $$ \bigl(\rho^2w''(\rho)+\rho w'(\rho)\bigr)\phi(t)=\rho^2 a^{- …

Let us look at the local problem: taking $X$ a non-zero smooth vector field in a neighborhood of 0 in $\mathbb R^3$, you may choose local coordinates such that $X=\partial_z$. If $π_1, π_2$ are smooth …

Let me try to reformulate (too long for a comment) your question by specializing it to a more particular (and more stable) case. Let $u_0:\mathbb R^2\rightarrow \mathbb R$, be a (smooth) Morse functio …