# explanation of (weak) null condition (as used in PDE)

I work in computational physics (numerical relativity), and I would like to better understand the null condition (as I believe we first introduced here), which helpful in determining the long time behavior of various PDEs that appear in geometric analysis. I have had trouble finding references online that are very accessible to someone with a more physics/computational background, and concisely introduce and summarize the ideas behind the null condition.

I would appreciate a concise explanation and motivation for the (weak) null condition, along with a (potentially heuristic) explanation of how it plays a role in proving long term existence for, e.g. various semilinear geometric PDE (as described, for example, in Klainerman's article linked above, or more recently as used here for the weak null condition). An explanation of how the condition applies to a (set of) simple example(s) would also be helpful. Either a description of this here, or a reference to literature suitable for someone without extensive PDE expertise would work!

I am no expert on this, but since nobody has left a comment/answer I will write what I know and give some references.

Restricting our attention to the wave equation, we want to consider the following initial value problem with small initial data:

$$\Box u = F(u,\partial u)$$ $$u(0,.) = \epsilon f$$ $$\partial_t u(0,.) = \epsilon g$$

Where $$f,g$$ are smooth and compactly supported functions in $$\mathbb{R}^3$$, $$\Box = -\partial_t^2 + \sum_i \partial_i^2$$ and $$F$$ is some nonlinearity depending in some ways on $$u$$ and its derivatives $$\partial u$$. The question is if we can determine a $$\textit{global}$$ (in time) solution to this system for some $$\epsilon > 0$$. Answering this question is hopelessly difficult, and in fact examples where $$F$$ has a fairly simple form have been shown to blow up in finite time. For example $$F_1 = (\partial_t u)^2$$ (John, F. Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29-51.)

The null condition is a restricition on the nonlinearity $$F$$ which (due to Klainerman) ensures global existence. If we instead consider $$F_2 = (\partial_t u)^2 - \sum_i (\partial_i u)^2$$ then it turns out that for $$F_2$$, the system above has global solution. The difference is that $$F_2$$ satisfies a special algebraic structure (the null condition) while $$F_1$$ does not.

The weak null condition (introduced by Lindblad and Rodnianski for their new proof of the stability of Minkowski space) is not as useful, in the sense that there is currently no similar results which guarentees global existence for a $$F$$ satisfying the weak null condition. However, all known examples where $$F$$ does satisfy the weak null condition do in fact possess global solution. A nice example is Einstein's equation (in wave coordinates). Interestingly, in dimension $$(n+1), n \geq 4$$ one can in fact prove general global existence (for small enough $$\epsilon$$), see Sogge's (reference below). The problematic dimension is $$3+1$$ which is relevant for relativity.

For a brief review of this stuff, chech out the last 6 pages of the lecutre notes.

The only other references that I know of are the ones from the PDE community, such as Sogge's book or Hörmander.

• Based on your reference to Ringstrom's lectures, is the null condition $F(u,\xi_i)=\mathcal{O}(u^3)+\mathcal{O}(\xi^3)$ for $\xi_i$ null with respect to the background geometry (for the system of PDE you wrote in your answer)? – PHY314 May 14 '19 at 14:37
• If you consider the coefficients $f_{\mu \nu}$ in the notes, then we easily see that e.g. $F_2$ above satisfies the null condition, since for this examples $f_{\mu \nu} = \eta_{\mu \nu}$, so that for any null vector $\xi$ it holds that $\xi^\mu \xi^\nu f_{\mu \nu} = 0$, so $F_2$ satisfies the null condition. – monolith28 May 15 '19 at 18:48