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It is sometimes referred to as "Glimm's existence theorem". Though in some ways the proof is "more famous" than the theorem, and is frequently referred to as "Glimm's difference scheme". Not that I …

It helps to consider components of $q$, in order to keep the notations clear. From what you have we can write $$ q_i'(t) = \int v_i \sum_{j} \nabla_{x_j} \phi \nabla_{v_j} f ~\mathrm{d}x ~\mathrm{d} …

Step 1: assuming scattering, there exists a solution $v$ to the linear Klein-Gordon equation such that $u-v \to 0$ in $H^1(\mathbb{R}^3)$ as $t \to \infty$. By Sobolev embedding this means that for an …

Fundamentally, the classical maximum principles for second order elliptic PDEs are based on the simple facts that The Hessian of a function at a local maximum is positive semidefinite. The full contr …

The example is almost trivial. For argument sake fix the number of spatial dimensions to be 3. If you consider radially symmetric solutions to the wave equation, one observes that if $$ u_{tt} = u_{ …

Why we do not study such estimates for hyperbolic equations? Because they are false. Now: you may ask "why are they false?" This is a fairly deep question, and answers often involve discussion of p …

Strichartz estimates on domains is a difficult problem! First: on bounded domains you cannot have any global in time Strichartz estimates. This is because of the presence of standing waves. (Set initi …

For an equation that is actually hyperbolic, this is well-known. Here are some classical references: Lars Garding, Cauchy's Problem for Hyperbolic Equations (1958) Jean Leray, Hyperbolic Differenti …

There is a typo in the paper. Look at the bottom two lines of page 22 which I transcribe here \begin{align} \|\phi\|_{L^3} & \leq \ldots \\ & \lesssim \mathscr{I}_0^{1/2} (1+t)^{1/2} ( \mathscr{I}_ …

Okay, so I would write your equations instead in the following form: $$ \partial_t u_i + v_i(t) \cdot \nabla u_i = b_i (t, \vec{x}, \vec{u}) $$ This is a system of transport equations and so can actua …

The problem is that your boundary conditions are incompatible with your equation. Your choice of boundary data implies that, were $u$ to be in $C^1([0,1]\times[0,1])$ (meaning that the derivative ex …

If you just want to have a lower-bound on the time of classical existence, it is not too hard, though the answer will be not as sharp as the case of the Burgers' equation. We will assume that $f_n$ …

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 …

The dynamic metric is the metric for the quasidiagonal system; and so for self consistency the solution can only be proven to exist (and be unique) when the metric (i.e. the solution itself) is hyper …

I just quickly read the proof you mentioned, and I think what is meant is following: Note that in Section 4.3 it is noted that any weak solution $U$ may be renormalized to be a continuous (in weak* t …