7

A small number of Strichartz estimates can be proven by virial methods, see the paper of Planchon and Vega at http://arxiv.org/abs/0712.4076 . Unfortunately, despite some effort, it does not appear that the methods cover the majority of Strichartz estimates. See also the interaction Morawetz inequalities, which are proven by the Morawetz multiplier method (...


7

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 equation, every $L^\infty$-data yields at least one bounded global-in-time "entropy" solution. This is done by Compensated Compactness (DiPerna 1983). However, ...


7

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 hyperbolic/Lorentzian. Noncompactness has zero impact whatsoever. Hyperbolic equations have finite speed of propagation and hence "local local uniqueness" (...


7

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 into linear factors over the reals, your equation is hyperbolic. However, "being hyperbolic" is never a complete sentence in the theory. One must say ...


6

First of all, your system seems to uncouple quite strongly. The first and third equations only involve the unknowns $v_1$ and $p_1$ and the second and fourth equations only involve $v_2$ and $p_2$, so you are really talking about two independent problems, so you should separate them this way. Second, if you add the first and third equations and set $z = ...


6

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 Differential Equations (Institute of Advanced Study, 1953): see especially Chapter VI, section 4 as well as the extensions in Part III. These are in addition to works of ...


6

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 $$S_0:= -\big(a^2-|x|^2\big)\ e_1\!\otimes e_1+y \otimes y,$$ and $$\lambda:={1\over2}\big(a^2-|x|^2\big).$$ This gives $\lambda$ as eigenvalue of $S$ of multiplicity $n-1$, with ...


6

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 convexity is not preserved by composition by the diffeomorphism $\phi$. In addition, if $n\ge3$, a generic quasi-linear system does not admit conservation ...


6

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 proof here. Theorem Let $\psi\in \mathscr{D}'(\mathbb{R}^{n+1})$ be a distributional solution of $\Box \psi = 0$. Suppose further that $\mathrm{supp}(\psi) \cap \...


6

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 actually be solved by using a variation of the Picard-Lindelof argument. (I am implicitly assuming that your function $b$ is suitably nice in certain ways, which are ...


5

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 $$E(t) = \int_{\{t\} \times \mathbb{R}^d} |\partial_t\phi|^2 + |\nabla \phi|^2 + m^2 \phi^2 ~\mathrm{d}x$$ then it can be shown that $E(t) = E(0)$ for every $t$. And hence if your "decay ...


5

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 equations: the space-time "ends" in finite proper time at the singularity, and there's not enough time (compare to the not-enough-space scenario you ...


5

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 answer is, in general, no. First, consider the case if you require $(\mathbb{R}^4,g)$ to be globally hyperbolic. Counterexample: Let $A$ be non-closed, ...


5

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 $p_k\in \mathbb{R}$ at points of a prescribed discrete subset $(x_k)_{k\ge1}$ of $\mathbb{R}_+$: $f'(x_k)=p_k$ for all $k\ge1$. Construction. We may assume w.l.o.g. $\omega$ is a ...


5

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 above equation becomes $$ \partial_tv=\partial_xw. $$ Being two different partial derivative of the same function $u$, $v$ and $w$ are related by the ...


5

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 any given $c > 1$. Therefore, if $M(0) > -1/c$, no blow-up occurs.


5

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 system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues ...


5

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 $at>0$).


5

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 found, the desired result follows from the representation formula (1.2) in the paper. This is an instance of a trace theorem, except in this case the argument is ...


5

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 (including remaining of Lorentzian signature) only in some open neighborhood of the Cauchy surface $\bar{M}$, without any guarantee that this neighborhood will be of ...


5

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 distributions that would be acting on smooth functions) is very well developed in Bär, Christian; Ginoux, Nicolas; Pfäffle, Frank, Wave equations on Lorentzian ...


4

Using your coordinates, we can define a Lorentzian metric $g = - dt^2 + \frac{1}{\alpha(t)} dx^2$. Then your equation takes the form \begin{equation} \square u + v^i \partial_i u + \gamma u = h , \end{equation} where $\square = \frac{1}{\sqrt{-\det g}} \partial_i \sqrt{-\det g} g^{ij} \partial_j$ is the (d'Alambert) wave operator for the metric $g$, $v^i \...


4

It may be that the bound is only fairly good for $r \ll n.$ This is supported by the following which is an elaboration on the comment by Boris Bukh. We can get a lower bound of $$ f(r,n)=\sum_{j=0}^n \binom{r}{j}.$$ This is the number of regions (all of which happen to be convex) in the complement of the union of $r$ hyperplanes in general position. This ...


4

You will find your answer in this article: http://www.sciencedirect.com/science/article/pii/0022247X89902059


4

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 appropriate implicit function theorem to prove existence theorems. Look up fully nonlinear elliptic PDEs for one well studied area.


4

When the RHS is 0, you are basically asking about the method of Riemann invariants. A quick summary of the method you can find in the second section of Lax (1964), J. Math. Phys. Alternatively, I am pretty sure it is also discussed somewhere in the second volume of Courant and Hilbert. EDIT: I see that there's also a decent write-up on Wikipedia, you ...


4

When $n=1$, you can always do this, at least near $t=0$, by solving a single inhomogeneous, linear first-order PDE; you can even arrange that $h_2 = h_1$. When $n>1$, there is a geometrical obstruction that can be computed in terms of $f_1$ and $\lambda g$. This is a classical fact in the geometry of PDE and characteristic systems. Here is a summary of ...


4

There is an extensive literature, this could be helpful entry point: Solving Navier-Stokes equations coupled with a heat transfer equation (2015) In this paper, the dynamics of an incompressible fluid in a bounded connected domain, described by Navier-Stokes equations coupled with a heat transfer equation, is investigated by a method inspired from ...


4

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) $$ \partial_t \int f^p dz = \int \partial_t f^p dz $$ Next, $$ 0 = \int \nabla_x \cdot (vf^p) dz $$ assuming $f$ decays suitably fast at infinity, and similarly $$ 0 = \int \nabla_v \cdot (Ff^p) ...


4

It helps to consider components of $q$, in order to keep the notations clear. From what you have we can write $$ q_i'(t) = \int v_i \sum_{j} \nabla_{x_j} \phi \nabla_{v_j} f ~\mathrm{d}x ~\mathrm{d}v $$ integrate by parts in $v$, and using that $\phi$ is independent of $v$ and $\nabla_{v_j} v_i = \delta_{ij}$ we get $$ q_i'(t) = - \int \nabla_{x_i} \phi ...


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