UC Berkeley

In this thesis, we consider two geometric, energy critical, semilinear wave equations arising from the wave maps problem (which is referred to as a nonlinear sigma model in many physics contexts) and Yang-Mills theory. The wave maps and Yang-Mills problems are families of equations, parameterized by various choices of domain and target manifolds, etc. The wave maps equation is geometric in the sense that the equation is independent of choice of coordinates on the domain and target manifolds. The Yang-Mills problem can be regarded as a geometric problem because it is invariant under gauge transformations. The specific symmetry reductions of these problems which we study are semilinear because their nonlinear terms involve only the unknown function, rather than its derivatives. Finally, both equations we consider admit a conserved energy and a scaling symmetry. The energy is invariant with respect to the scaling symmetry in the dimensions in which we study each equation, which is why the equations are said to be energy critical. We will now describe the equations considered in more detail. \

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The wave maps problem is an extension of the scalar linear wave equation to the case where the unknown function maps a Lorentzian manifold into a Riemannian manifold, $(M,g)$. The wave maps equation can be derived from the least action principle applied to the following natural extension of the usual wave equation action.

$$\mathcal{S}(\Phi) = \int_{\mathbb{R}^{1+d}} \langle \partial_{\alpha} \Phi(t,x),\partial^{\alpha} \Phi(t,x) \rangle_{g(\Phi(t,x))} dtdx$$

where the $\alpha$ indices are contracted using the Minkowski metric, which we take to be

$$m=\text{diag}(-1,1,1,\dots,1)$$ Now, we will describe the particular choice of target manifold and symmetry reductions considered. First of all, we will focus on the case $(M,g) = (\mathbb{S}^{2},\mathring{g})$, where $\mathring{g}$ denotes the usual round metric on $\mathbb{S}^{2}$. We will regard $\Phi$ as a map into $\mathbb{R}^{3}$ with unit Euclidean norm. The wave maps equation then becomes

$$-\partial_{\alpha}\partial^{\alpha} \Phi(t,x) = \Box_{\mathbb{R}^{1+d}} \Phi(t,x) = \Phi(t,x) \left(\partial^{\alpha}\Phi(t,x) \cdot \partial_{\alpha}\Phi(t,x)\right), \quad \Phi(t,x)\cdot \Phi(t,x)=1$$

where $\cdot$ is the Euclidean inner product on $\mathbb{R}^{3}$. We then make a symmetry reduction of the problem, and consider solutions which are 1-equivariant by using polar coordinates $(r,\varphi)$ on $\mathbb{R}^{2}$, and writing

$$\Phi_{u}(t,r,\varphi) = \left(

\begin{array}{c}

\cos(\varphi) \sin(u(t,r)) \

\sin(\varphi)\sin(u(t,r)) \

\cos(u(t,r)) \

\end{array}

\right)$$

Then, the wave maps equation reduces to the following semilinear wave equation

$$-\partial_{tt}u+\partial_{rr}u+\frac{1}{r}\partial_{r}u-\frac{\sin(2u)}{2r^{2}}=0$$

which has the following conserved energy

$$E_{WM}(u,\partial_{t}u)=\pi \int_{0}^{\infty} \left(\left(\partial_{t}u\right)^{2}+\frac{\sin^{2}(u)}{r^{2}} + \left(\partial_{r}u\right)^{2}\right) r dr$$

Note that the energy is invariant if we replace $u$ by $u_{\lambda}$ defined by $u_{\lambda}(t,r) = u(\lambda t, \lambda r)$, for $\lambda>0$. Such a scaling transformation is also a symmetry of the wave maps equation above.\

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Given a Lie Group $G$, the (free) Yang-Mills equation we will consider is an equation for a $Lie(G)$-valued one-form $A$ defined on $\mathbb{R}^{1+d}$. We consider the Yang-Mills equation in $1+4$ dimensions, with gauge group $SO(4)$. Therefore, $A$ (which is sometimes called the gauge field) is a $\text{Lie}(SO(4))$-valued one-form on $\mathbb{R}^{1+4}$. We write $A=A_{\mu} dx^{\mu}$, where, for each $\mu$, $A_{\mu}$ is a $\text{Lie}(SO(4))$-valued function, defined on $\mathbb{R}^{1+4}$. Defining $F$, a $\text{Lie}(SO(4))$-valued two-form on $\mathbb{R}^{1+4}$ by

$$F=\frac{1}{2}F_{\mu\nu} dx^{\mu} \wedge dx^{\nu}, \quad F_{\mu\nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}]$$

the Yang-Mills equation can be written as

$$-\partial_{t}F_{0\nu}-[A_{0},F_{0\nu}] + \sum_{\mu=1}^{4}\left(\partial_{\mu} F_{\mu \nu} + [A_{\mu},F_{\mu\nu}]\right)=0, \quad \text{ for } \nu=0,1,2,3,4$$

where $0$ on the right-hand is the zero in $\text{Lie}(SO(4))$.\

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The Yang-Mills equation is also invariant under gauge transformations, which are transformations of $A$ of the form

$$A_{\mu} \rightarrow g A_{\mu} g^{-1} - \partial_{\mu}g g^{-1}$$

where $g: \mathbb{R}^{1+4} \rightarrow SO(4)$.\

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We make the equivariant ansatz (which has also been studied in prior works)

$$A_{\mu}^{i,j}(t,x) = \left(\delta^{i}_{\mu}x^{j}-\delta^{j}_{\mu}x^{i}\right)\left(\frac{u(t,|x|)-1}{|x|^{2}}\right), \quad 0 \leq \mu \leq 4, \quad 1 \leq i,j \leq 4$$

and the Yang-Mills equation reduces to a single semilinear wave equation

\begin{equation}-\partial_{tt}u+\partial_{rr}u+\frac{1}{r}\partial_{r}u+\frac{2u(1-u^{2})}{r^{2}}=0\end{equation}

This wave equation conserves the following energy

\begin{equation} E_{YM}(u,\partial_{t}u) = \frac{1}{2} \int_{0}^{\infty} \left(\left(\partial_{t}u\right)^{2}+\left(\partial_{r}u\right)^{2}+\frac{(1-u^{2})^{2}}{r^{2}}\right) r dr\end{equation}

which is invariant under the scaling symmetry:

$$u \rightarrow u_{\lambda}, \quad \text{ where } u_{\lambda}(t,r) = u(\lambda t, \lambda r)$$

Each equation mentioned above admits time-independent, smooth solutions, with localized energy density, called solitons. For the wave maps problem considered above, the soliton is given by

$$Q_{1}(r) = 2 \arctan(r)$$

For the Yang-Mills problem, the soliton is

$$Q_{1}(r) = \frac{1-r^{2}}{1+r^{2}}$$

By applying the aforementioned scaling symmetry to $Q_{1}$, one obtains a family of soliton solutions, $Q_{\lambda}$ for $\lambda> 0$, given by

$$Q_{\lambda}(r)=Q(r\lambda)$$

As mentioned earlier, we consider energy critical equations, so all $Q_{\lambda}$ have the same energy.\

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In this thesis, we study global in time soliton dynamics for single solitons coupled to radiation. More precisely, we construct globally defined solutions, $u$, which can be decomposed as follows.

\begin{equation}\label{solutiondecomp}u(t,r)=Q_{\frac{1}{\lambda(t)}}(r)+f(t,r)+v_{e}(t,r)\end{equation}

The function $f$ represents radiation coupled to the soliton with time-dependent scale, and is a solution to the following linear wave equation

\begin{equation}\begin{split}&-\partial_{tt}f+\partial_{rr}f+\frac{1}{r}\partial_{r}f-\frac{f}{r^{2}}=0, \quad \text{ for wave maps}\

&-\partial_{tt}f+\partial_{rr}f+\frac{1}{r}\partial_{r}f-\frac{4}{r^{2}}f=0, \quad \text{ for Yang-Mills}\end{split}\end{equation}

(In the main body of the thesis, $f$ will be denoted as $v_{2}$ in the wave maps work, but as $v_{1}$ in the Yang-Mills work). The function $v_{e}$ appearing in \eqref{solutiondecomp} is a correction which is small in an appropriate sense as time approaches infinity. We also provide a precise relation between the asymptotics of the time-dependent soliton length scale, $\lambda(t)$, and the coupled radiation $f$.\

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For the wave maps equation, for any choice of function $\lambda_{0}(t)$ in an appropriate symbol class, we construct a solution as described above, with $\lambda(t)$ asymptotically equal to $\lambda_{0}(t)$. Some examples of the $\lambda_{0}$ in our symbol class are

$$\lambda_{0}(t) = \frac{1}{\log^{b}(t)}, \quad \lambda_{0}(t) = \frac{2+\sin(\log(\log(t)))}{\log^{b}(t)}, \quad t \gg 1$$

For the Yang-Mills equation mentioned above, we construct a class of solutions as in \eqref{solutiondecomp}. The main difference between the wave maps result, and the result here is that $\lambda(t)$ is asymptotically constant for all of our solutions to the Yang-Mills problem. This is true, even though our set of solutions includes ones for which the radiation $f$ in \eqref{solutiondecomp} can be quite large, and in fact ``logarithmically'' close to having infinite energy. In fact, in the setup of this work, the soliton length scale asymptoting to a constant is a necessary condition for the radiation $f$ to have finite energy. Another interesting point of this construction is that, for each choice of $f$ in our admissible class of functions, there exists a one-parameter family of solutions as in \eqref{solutiondecomp} with $\lambda(t)$ having any asymptotic value.