User:Nikodem0/Stability of nonlinear wave equations

This page describes a project on stability of nonlinear wave equations of various types, from simple (toy models) to realistic (Einstein eqs.).

Nonlinear wave equations
$$\Box u = F(u,r)$$ [Yang-Mills in sph.sym.; wave maps (?->?) with sym. Ansatz]
 * Semilinear with no derivatives

$$\Box u + V(r) u = F(u,r)$$ Are there stable static solutions of the above equation?
 * Semilinear with no derivatives around a nontrivial "background" solution ($$u\neq 0$$) they get a potential term

$$\Box_g u = F(u,r)$$ It takes the form $$g^{\mu\nu}\partial_\mu \partial_\nu u + f^\mu \partial_\mu u = F(u,r)$$ with $$f^\mu := g^{\alpha\beta} \Gamma^{\lambda}_{\alpha\beta} = g^{\mu\nu} \partial_\nu \ln\sqrt{|g|}$$
 * Semilinear with no derivatives on a curved spacetime

$$\Box_{g(u)} u = F(u,r)$$ which takes the form $$g^{\mu\nu}(u)\partial_\mu \partial_\nu u + f^\mu(u) \partial_\mu u = F(u,r)$$
 * Quasilinear with no derivatives on a dynamical curved spacetime ("self-gravitating")

$$\Box u = Q(\partial u, \partial u) = q^{\alpha \beta}(u) \partial_\alpha u \partial_\beta u $$
 * Semilinear with quadratic derivatives

with the null structure: $$q^{ab} v_a v_b=0$$ for all null: $$\eta^{ab} v_a v_b=0$$.

$$\Box u + f^\mu \partial_\mu u = Q(u, u) = q^{\alpha \beta}(u) \partial_\alpha u \partial_\beta u $$
 * Semilinear with quadratic derivatives around a nontrivial "background" solution ($$u\neq 0$$) they get linear terms

The potential term and terms linear in derivatives cause decay problems. How to control them?

$$\Box_g u = Q(\partial u, \partial u) = q^{\alpha \beta}(u) \partial_\alpha u \partial_\beta u $$ with the null structure: $$q^{ab} v_a v_b=0$$ for all null (w.r.t the background): $$g^{ab} v_a v_b=0$$.
 * Semilinear with quadratic derivatives on a curved spacetime

$$\Box_{g(u)} u = Q(\partial u, \partial u) = q^{\alpha \beta}(u) \partial_\alpha u \partial_\beta u $$
 * Quasilinear with quadratic derivatives on a dynamical curved spacetime ("self-gravitating")

Systems of nonlinear wave equations
$$\Box u^A = Q^A(\partial u^B, \partial u^C) = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C $$
 * System of wave equations

with null/weak null/polarized null condition. In particular, interesting is the case appearing in study of the Einstein eqs.

$$\Box g_{ab} = Q^A(\partial g, \partial g) := S_{ab}(\partial g, \partial g) := p_{cdef} \partial_a g^{cd} \partial_b g^{ef}$$

without the null-condition.

$$\Box u^A + {f^A_B}^\mu \partial_\mu u^B + V^A_B u^B = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C $$ [sph.sym. gravitating scalar field on Minkowski or Schwarzschild; triaxial Bianchi IX?]
 * System of wave equations around a nontrivial "background" solution

$$\Box_g u^A = Q^A(\partial u^B, \partial u^C) = q^A_{BC}{}^{\alpha \beta}(u) \partial_\alpha u^B \partial_\beta u^C $$ If $$g=g(u)$$ then it gives a system of quasilinear wave equations [triaxial Bianchi IX?]
 * System of wave equations on a nontrivial background

Again, the above situation without the null-condition is especially interesting.

The main questions are:

1) Stability and decay around a static "vacuum" (u=0)

1a) in presence of the potential term

1b) in presence of the linear terms

2) Stability and decay around a nontrivial static background ($$u\neq 0$$)

2a) in presence of the potential and/or the linear terms