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What you've found is basically "survivor bias", so it helps for me to describe a bit where the null conditions came about. Assertion 1: Quasilinear partial differential equations, in general, are too …

For simplicity I'll assume $u$ is scalar valued, but I am pretty sure the discussion also works for $u$ that is a section of some vector bundle over $M$ (if the wave operator is quasidiagonal). Additi …

I don't think in the level of generality you are looking at you can say anything useful. Let me give two examples with very contrasting behaviors. Here I am, per your comment, allowing myself to think …

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 …

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 …

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 …

By replacing $\phi$ by $-\phi$ you can set $\alpha$ to be positive (or negative, if you wish). By replacing $\phi$ by $\lambda \phi$ you can rescale away $\alpha$. So you can set $\alpha$ to be either …

You can take a look at Herbert Koch's contribution in Koch, Herbert; Tataru, Daniel; Vişan, Monica, Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave …

You cannot. Let $v$ be a Dirichlet eigenfunction of $-\Delta$ on the domain $\Omega$ with eigenvalue $\lambda > 0$. The function $$ u(t,x) = \sin(\sqrt{\lambda}t) v(x) $$ solves the wave equation. The …

$$ \tilde{u}_{tt} - \frac{2}{t}\tilde{u}_{t}-\Delta \tilde{u} = g_t -\frac{2}{t^2}\tilde{u} $$ $$ \dot{E} = 2 \int \tilde{u}_t (2 t^{-1} \tilde{u}_t + g_t - 2 t^{-2} \tilde{u} ) $$ $$ \dot{E} = 2 \int …

Define using $H(t) = \int (u_t)^2 + |\nabla u|^2 ~dx $ the standard energy. Taking the time derivative you find $$ \frac{d}{dt}H(t) = 2 \int u_t( g + \frac{2}{t} u_t)$$ Writing $\|\cdot \|$ for the $L …

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 …

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 …

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 key is to estimate the integral (where $g\in C^\infty_c(\mathbb{R}^3)$ by assumption) $$\tag{A} \iint_{|x - y| = t} g(y)~ dS_y \lesssim \iiint_{|x-y| \geq t} |D g|~dy. $$ Once this estimate is fou …