Generalized Virial Identity for NLS

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=Example 2=
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[[Category:Concept]]

Latest revision as of 16:15, 27 May 2010

Linking back to the course: 2010 MAT495H1 Group: Solitons and Blowup. (page under development)

Contents

The Generalized Nonlinear Schrodinger

We consider the Cauchy problem for the Generalized Nonlinear Schrodinger equation

$$(i\partial_t + \Delta)u =\pm F'(|u|^{2})u$$ with initial condition $$u(0,x)=u_{o}(x).$$ We have that $F'(x)\ge 0$. In the focusing case we have a $+$ sign and defocusing a $-$ sign. This can be specialized into several specific NLS equations, but we will use this general case to note the conserved quantities.

Conserved Quantities

We can show that mass, momentum, and energy are conserved. Specifically, $$\frac{\partial}{\partial t}\int_{\mathbb{R}^{n}}|u(t,x)|^{2}dx=0,$$

$$\frac{\partial}{\partial t}2\Im\int_{\mathbb{R}^{n}} \bar{u}(t)\nabla u(t)dx=0,$$

and

$$\frac{\partial}{\partial t}H[u(t)]=\frac{\partial}{\partial t}\frac{1}{2}\int_{\mathbb{R}^{n}} |\nabla u(t)|^{2}\pm F(|u(t)|^{2})=0.$$

We will not show these conserved quantities but they are a simple manipulations of the NLS equation. We introduce some notation and physical interpretation to make our the rest of what is written easier to read and understand. Imagine the solution $u$ as a particle, the mass as $$T_{oo}:=|u|^{2},$$ the momentum as $$T_{oj}:=T_{jo}=2\Im (u\bar{u}_j),$$ and the momentum current as

$$T_{jk}:=T_{kj}=4\Re(u_k\bar{u}_j)-\delta_{j,k}\Delta(|u|^{2})+2\delta_{j,k}\sigma(|u|^2).$$ Where $\sigma(z):=zF'(z)-F(z)\sim F(z)$.

From these definitions, one can show by computation that $$\partial_{t}T_{oo}=\partial_{x_{j}}T_{oj}$$ and $$\partial_{t}T_{ko}=\partial_{x_{j}}T_{kj}.$$ Note, we use Einstein notation here.

The first equation comes from

\begin{align} \partial_t (u\bar{u})&=u_t\bar{u}+u\bar{u}_t\\ &=-i(iu_t\bar{u})+c.c.\\ &=-i(-\Delta u+F'(|u|^2)u)\bar{u}+c.c.\\ &=i(\Delta u)\bar{u}+c.c. \end{align}

and

\begin{align} \partial_j[2Im(u \bar{u}_j)]&=(-i)\partial_j(u\bar{u}_j-\bar{u}u_j)\\ &=(-i)(u_j\bar{u}_j+u\bar{u}_{jj}-\bar{u}_ju_j-\bar{u}u_{jj})\\ &=i(\bar{u}u_{jj}-u\bar{u}_{jj}) \end{align}

where $c.c.$ means complex conjuate of the previous terms.

The LHS of the second equation is

\begin{align} \partial_t[2\Im(u\bar{u}_k)]&=(-i)\partial_t[u\bar{u}_k-c.c]\\ &=(-i)[u_t\bar{u}_k+u\bar{u}_{kt}-c.c.]\\ &=-[(iu_t)\bar{u}_k-u\overline{(iu)}_{kt}]+c.c.\\ &=-[(-\Delta u+F'(|u|^2)u)\bar{u_k}-u\overline{(-\Delta u+F'(|u|^2)u)}_k]+c.c.\\ &=(\Delta u)\bar{u}_k-(\Delta\bar{u}_k)u-F'(|u|^2)u\bar{u}_k+[F'(|u|^2)\bar{u}]_ku+c.c. \end{align}

and because


$$\Delta(|u|^2)_k=[(\Delta u)\bar{u}+2\nabla u\cdot\nabla\bar{u}+u(\Delta\bar{u})]_k,$$

the RHS of the second equation is

\begin{align} \partial_j T_{kj}&=\partial_j[2(u_k\bar{u}_j+\bar{u}_k u_j)-\delta_{jk}\Delta(|u|^2)+2\delta_{jk}\sigma(|u|^2)]\\ &=(\Delta\bar{u})u_k+(\Delta u)\bar{u}_k-(\Delta u_k)\bar{u}-u(\Delta \bar{u}_k)+[|u|^2F'(|u|^2)-F(|u|^2)]_k, \end{align}

hence the second equation holds.

Generalized Virial Identity

We now introduce the the the Virial potential associated to $a$, $$V_{o}^{a}(t):=\int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx=\int_{\mathbb{R}^{d}} a(t,x) T_{oo}dx,$$ and the Morawetz action $$M_{o}^{a}(t):=\partial_t V_{o}^{a}(t)=\int_{\mathbb{R}^{d}} a_{t}T_{oo}+a\partial_{x_{j}}T_{oj}dx=\int_{\mathbb{R}^{d}} a_{t}T_{oo}-a_{x_{j}}T_{oj}dx.$$ The virial identity is derived from the two equalities show in the previous section.

\begin{align} \partial_t^2V_0^a(t)=\partial_tM_0^a&=\int_{\mathbb{R}^{d}} a_{tt}T_{00t}+a_tT_{00} dx-\int_{\mathbb{R}^{d}}a_{tj}T_{0j}+a_jT_{0jt}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2+a_{t}\partial_jT_{0j}dx-\int_{\mathbb{R}^{d}}a_{tj}T_{0j}+a_j\partial_kT_{jk}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2dx-\int_{\mathbb{R}^{d}}2a_{tj}T_{0j}+a_j\partial_kT_{jk}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2dx-4\int_{\mathbb{R}^{d}}a_{tj}\Im(u\bar{u}_j)-a_{jk}T_{jk}dx\\ &=\int_{\mathbb{R}^{d}}(a_{tt}-\Delta\Delta a)|u|^2dx-4\int_{\mathbb{R}^{d}}a_{tj}\Im(u\bar{u}_j)dx+4\int_{\mathbb{R}^{d}} a_{jk}\Re(u_k\bar{u}_j)dx\\ & +2\int_{\mathbb{R}^{d}}\Delta a\sigma(|u|^2)dx \end{align}

Example One: Virial Identity

If we take the weight $a(x)=|x|^{2}$, $a_{jk}=2\delta_{jk}$, $\nabla a=2d$, $\nabla\nabla a=0$, to arrive at $$\partial_{t}^{2}V_{o}^{a}(t)=+4\int_{\mathbb{R}^{d}} \delta_{jk}\Re(u_k\bar{u}_j)dx+4d\int_{\mathbb{R}^{d}}\sigma(|u|^2)dx$$ $$=8H+(\le 0)\le8h<0$$ since we are assuming the focusing case and specific initial data with negative initial (and therefore all-time) energy. We have shown $$\partial_{t}^{2} \int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx<0$$ which tells us that $\int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx$ is positive and monotone decreasing, telling us that at the point at which this equals zero, say $t=T^\star$ we have nonexistance of solution. $T^\star$ is the blow up time in this case. This is the argument which is found in Glassey's paper.

Example 2

To follow.

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