User:Nikodem0/Models of nonlinear wave equations

= Semilinear (without derivatives) =

$$\Box u + V u = F(u,r)$$

Pointwise decay [Jojn, Asakura, Strauss, NS]: for $$|V(x)|\lesssim \langle x\rangle^{-k}$$ and $$|F(u)|\lesssim u^p$$, $$|F(u)-F(v)|\lesssim |u|^{p-1} |u-v|$$ it holds $$ $$
 * u(t,x)| \leq \frac{C}{\langle t+|x|\rangle \langle t-|x|\rangle^{q-1}},\qquad q:=\min(p-1,k,m)

Perturbation theory: ... (almost done N+PB)

on Schwarzschild background
Static, spherically symmetric background $$\rightarrow$$ tortoise coordinate $$r_* = r+ \log(r-2M) \cong r$$ for $$r_*\rightarrow +\infty$$ $$ \partial_t^2 u_l - \partial_{r_*}^2 u_l + \frac{l(l+1)}{r^2} u_l + V(r_*) u_l= ... $$ But here $$r_*\in (-\infty,+\infty)$$!

= Semilinear (with derivatives) =

Wave maps
$$\Box u = F(u,r) Q(\partial u, \partial u)$$ $$M^{1+3}\rightarrow S^3$$ with Ansatz: spherical symmetry, corotational $$ \partial_t^2 u - \partial_r^2 u -\frac{2}{r} \partial_r u = -\frac{\sin(2u)}{r^2} \cong -\frac{2}{r^2}u + \frac{4u^3}{3r^2} + O(u^5) $$

Stability? Perturbations decay like $$t^{-5}$$ [~Bizon-NUM]

-Other (general)...? Self-gravitating?

Yang-Mills
$$M^{1+3}\rightarrow SU(2)$$ with spherical symmetry $$ \partial_t^2 u - \partial_r^2 u -\frac{2}{r} \partial_r u + \frac{2}{r^2} u = -u^3 - \frac{3u^2}{r} $$

Stability [Eardley & Moncrief]. Decay [...] -- optimal rate? Perturbations decay like $$t^{-4}$$ [Bizon-NUM] (follows from Christodoulou's conformal technique)

= Quasilinear with null condition =

Skyrme
$$M^{1+3}\rightarrow S^3$$ with Ansatz: spherical symmetry, corotational $$ (\rho \dot{u})\dot{} - (\rho u')' = -\sin(2u) -\alpha^2 \sin(2u) \left(\frac{\sin^2(u)}{r^2} + u'^2 - \dot{u}^2\right) $$ where $$\rho \equiv r^2 + 2 \alpha^2 \sin^2 u$$. RHS has null structure.


 * Around $$u\equiv 0$$: null structure

Stability and decay? (follows from general theorems for quasilin.?)


 * Has nontrivial topological sectors. $$u_1$$ is linearly stable (even against nonradial perturbations). Perturbations around $$u_1$$: null structure with linear derivatives

Stability - not known. Perturbations decay like $$t^{-5}$$ (linear $$t^{-8}$$) [Bizon-NUM]

solitons"]
 * Static solutions $$u_k$$ with $$k>1$$ are linearly unstable. In sector $$k=2$$ the static energy minimizer is axially symmetric. [see Manton "Topological

-General (no Ansatz) ?

= Quasilinear systems with null condition =

5D Bianchi IX (triaxial)
$$ \Box u^A = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C + h^A(u) $$ with $$q^A_{BC}{}^{\alpha \beta}(u)$$ satisfying the null condition.

-Orbital stability (in mass param.) [Dafermos & Holzegel]; no decay info.

$$\Box u^A + {f^A_B}^\mu \partial_\mu u^B + V^A_B u^B = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C + h^A(u)$$ where $$q^A_{BC}{}^{\alpha \beta}(u)$$ satisfies the null condition.
 * Around Minkowski or Schwarzschild: system of wave equations with linear terms

Einstein + scalar field / spherical symmetry
$$ \Box u^A = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C + h^A(u) $$ with $$q^A_{BC}{}^{\alpha \beta}(u)$$ satisfying the null condition.

-stablity and decay [Christodoulou] (???)


 * Around Minkowski or Schwarzschild: system of wave equations with linear terms

$$\Box u^A + {f^A_B}^\mu \partial_\mu u^B + V^A_B u^B = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C + h^A(u)$$ with $$q^A_{BC}{}^{\alpha \beta}(u)$$ satisfying the null condition.

= Quasilinear systems with weak null condition =

Einstein eqs (+ vacuum)

 * Around Minkowski

Stability and decay [Christodoulou-Klainerman, Lindblad-Rodnianski]


 * Around Schwarzschild

No stability, no decay results.

Einstein + scalar field OR electromagnetic field ???

 * Around Minkowski?


 * Around Reissner-Nordstrøm?