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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 …

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 …

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 …

Since I haven't been able to track down Selberg's lecture notes since he moved to Bergen, and since the proof of the result I mentioned in this comment is super-short anyway, let me just include a pro …

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 …

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 …

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 …

A longish bunch of remarks: there's a big leap that doesn't make sense between your question and your motivation. On the maximally extended Schwawrzschild solution there is no decay to the wave equat …

There are a few things that can be said about this, and it depends on how one "approaches" infinity. Wave equations satisfy conservation of energy. Let $$E(t) = \int_{\{t\} \times \mathbb{R}^d} |\par …

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 …

Your question can be equivalently phrased as: Given a closed two-form $F$ on $\mathbb{R}^4$, is it always possible to find a Lorentzian metric on $\mathbb{R}^4$ such that $\delta F = 0$. Then t …

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 …

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 …

If $q$ is not signed, then in general the solution need not be unique. The question of uniqueness can be reduced to the case where $f \equiv 0$. In this case, the constant $0$ function obviously solve …