28
votes
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Why don't we study hyperbolic equations as elliptic and parabolic equations?
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 ...
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13
votes
Why don't we study hyperbolic equations as elliptic and parabolic equations?
For hyperbolic PDE's, we have two aspects that distinguish them from elliptic and parabolic PDE's and thus forbid one from being able to obtain estimates in the form you want:
Usually the loss of ...
- 7,817
8
votes
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Looking for references to study $U^p$ and $V^p$ spaces
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 ...
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8
votes
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Maximum principle and linear transport
This is not a transport equation. It is a conservation law. The difference between these class is that a TE is of the form $\partial_tu+a(t,x)\cdot\nabla_xu=0$, for which the essential supremum/...
- 52.3k
7
votes
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BV functions and wave equation
The answer to this question depends a lot on the space dimension $n$. It is true that if $n=1$, the Cauchy problem has been studied with data in either $L^\infty(R)$ or $BV(R)$. For superlinear wave ...
- 52.3k
7
votes
Definition of a system being hyperbolic
They are not. First of all, the existence of a convex entropy is not meaningful for a system given in this quasi-linear form. The reason is that you might make a change $v=\phi(u)$ of unknown, but the ...
- 52.3k
7
votes
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Uniqueness of solution of the wave equation
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 ...
- 39k
7
votes
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Preservation of metric signature in Cauchy problem for the Einstein equations
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 ...
- 39k
7
votes
Is this equation of hyperbolic type?
You made a confusion between the symbol (here $-(1+\xi^2)\tau^2+\xi^4$), and the principal symbol, which gathers the monomials of highest degree. Since the latter is $\xi^4-\xi^2\tau^2$, which splits ...
- 52.3k
7
votes
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A certain solution for Sine-Gordon Equation
Let me start by writing $1/U$ instead of $U$. The case when $U$ or $1/U$ vanishes (or $V$ vanishes) would require special handling of course. Then your constraint is
$$ \frac{\omega_u}{\omega_v} = \...
- 21.5k
6
votes
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Finite speed propagation by finite energy method
For an equation that is actually hyperbolic, this is well-known. Here are some classical references:
Lars Garding, Cauchy's Problem for Hyperbolic Equations (1958)
Jean Leray, Hyperbolic ...
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6
votes
Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation
A prototypical example: the wave equation (for sake of simplicity let's work in 1d, but the idea is general):
$$ \partial_t^2u=\partial_x^2u. $$
Let $v=\partial_tu$ and $w=\partial_xu$. Then the ...
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6
votes
A curious determinant of quadratic forms
Let's put $x:=(0,X)\in k^{n+1}$ and $y:=-ae_1+x\in k^{n+1}$. Then $S$ writes as a symmetric rank-$2$ perturbation of a multiple of the identity, $S=S_0+ \lambda I_{n+1},$ with
$$S_0:= -\big(a^2-|x|^...
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6
votes
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Existence of a solution for a quasilinear hyperbolic system of PDEs with many state variables
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 ...
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6
votes
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The determinant as a differential operator
Gårding's differential operator (introduced in Extension of a Formula by Cayley to Symmetric Determinants) is discussed by Turnbull in Symmetric Determinants and the Cayley and Capelli Operator:
The ...
- 188k
6
votes
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Closed-form solution to hyperbolic PDE
This is known as the Goursat problem, because the boundary condition are given over two intersecting characteristic lines. Notice the necessary condition (which turns out to be sufficient):
$$c_1(0)=...
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6
votes
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Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but ...
- 108k
6
votes
Does there exist an electromagnetic analogue of Einstein's field equations?
There is a profound conceptual shift between Gravitoelectromagnetism and General Relativity that cannot analogously occur between Maxwell's equations and any "???".
In General Relativity, ...
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6
votes
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Quasilinear wave equations without (weak) null conditions and conjectures
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 ...
- 39k
5
votes
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Principal symbol for non-linear differential operators
I've seen only the first. It is indeed used mostly for identifying whether a nonlinear PDE is elliptic, hyperbolic, or parabolic. If so, one can use the respective linear theory, along with the ...
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5
votes
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Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions
More generally: if $\omega$ is a modulus of continuity with $\omega'(0)=\infty$ there is an $\omega$-continuous, smooth function $f$ on $\mathbb{R}_+$, with prescribed derivative $p_k\in \mathbb{R}$ ...
- 60.5k
5
votes
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A problem about closed 2-forms on Minkowski space
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 the ...
- 39k
5
votes
Is this a "contradiction" on stochastic Burgers' equation? How to understand it?
It is not true that the bound $dM/dt = -\beta M^2$ implies that $M$ blows up almost surely. For example, with $b = 2$, there is a non-zero probability that $\beta < ce^{-t}$ for all $t>0$, for ...
- 10.3k
5
votes
Definition of a system being hyperbolic
No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the ...
- 188k
5
votes
About Dirac function
Your notion and the equality in the book are consistent, since
$$
\delta (a^2 t^2 - |x|^2 ) = \frac{1}{2|at|} (\delta (at-|x|) + \delta (at+|x|) )
$$
(I suppose an assumption is being made that $at>...
- 6,696
5
votes
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Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The construction of adding $p$ as an additional element to the vector $\mathbf u$ only hides it in the vector $\mathbf v$, without actually eliminating it from the Euler equation. This can be easily ...
- 188k
5
votes
Decay estimate on wave equation
The key is to estimate the integral (where $g\in C^\infty_c(\mathbb{R}^3)$ by assumption)
$$\tag{A} \iint_{|x - y| = t} g(y)~ dS_y \lesssim \iiint_{|x-y| \geq t} |D g|~dy. $$
Once this estimate is ...
- 39k
5
votes
Preservation of metric signature in Cauchy problem for the Einstein equations
I will add a pessimistic answer. You are right that Choque-Bruhat's (and any related local-in-time) existence result only guarantees that the solution metric exists and is sufficiently regular (...
- 21.5k
5
votes
Accepted
spaces of smooth functions for linear hyperbolic PDE
For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as ...
- 21.5k
5
votes
Accepted
On a nonlinear wave equation
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 ...
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