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Questions about partial differential equations of hyperbolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

2 votes
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How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\...

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 $ …
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1 vote

Difference between semilinear and fully nonlinear

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 …
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1 vote

Asymptotics for solution of transport equation and characteristics

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 …
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2 votes

Uniqueness of solution of the wave equation

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 …
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2 votes

Non-trivial examples of regular Lagrangian flow in BV case

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 …
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1 vote

Banach space-valued test functions in the definition of a weak solution of a PDE problem

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 …
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-1 votes
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2D wave equation with gaussian boundary condition

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^{- …
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2 votes
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Boundary behaviour of a second order pde with characteristics

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 …
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0 votes

Structure of sign changes under the heat flow

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 …
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