Celebratio Mathematica

Cast a stone in­to a still lake. There is a large splash, and waves be­gin ra­di­at­ing out from the splash point on the sur­face of the wa­ter. But, as time passes, the amp­litude of the waves de­cays to zero.

This type of be­ha­vi­or is com­mon in phys­ic­al waves, and also in the par­tial dif­fer­en­tial equa­tions used in math­em­at­ics to mod­el these waves. Let us be­gin with the clas­sic­al wave equa­tion $$\begin{array}{}\text{(1)}& -{\partial }_{tt}u+\mathrm{\Delta }u=0,\end{array}$$ where $u:\mathbb{R}\times {\mathbb{R}}^3\to \mathbb{R}$ is a func­tion of both time $t\in \mathbb{R}$ and space $x\in {\mathbb{R}}^3$, which is a simple mod­el for the amp­litude of a wave propagat­ing at unit speed in three-di­men­sion­al space; here $$\mathrm{\Delta }=\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\frac{{\partial }^2}{\partial x_3^2}$$ de­notes the spa­tial Lapla­cian. One can veri­fy that one has the fam­ily of ex­pli­cit solu­tions $$\begin{array}{}\text{(2)}& u(t,x)=\frac{F(t+|x|)-F(t-|x|)}{|x|}\end{array}$$ to $\text{(1)}$ for any smooth, com­pactly sup­por­ted func­tion $F:\mathbb{R}\to \mathbb{R}$, where $$|x|=\sqrt{x_1^2+x_2^2+x_3^2}$$ de­notes the Eu­c­lidean mag­nitude of a po­s­i­tion $x\in {\mathbb{R}}^3$. The dis­pers­ive nature of this equa­tion can be seen in the ob­ser­va­tion that the amp­litude $$\underset{x\in {\mathbb{R}}^3}{sup}|u(t,x)|$$ of such solu­tions de­cays to zero as $t\to \pm \mathrm{\infty }$, whilst oth­er quant­it­ies such as the en­ergy $${\int }_{{\mathbb{R}}^3}\frac{1}{2}|{\partial }_{t}u(t,x){|}^2+\frac{1}{2}|\mathrm{\nabla }u(t,x){|}^{2}dx$$ stay con­stant in time (and in par­tic­u­lar do not de­cay to zero).

The wave equa­tion can be viewed as a spe­cial case of the more gen­er­al lin­ear Klein–Gor­don equa­tion $$\begin{array}{}\text{(3)}& -{\partial }_{tt}u+\mathrm{\Delta }u=m^{2}u,\end{array}$$ where $m\ge 0$ is a con­stant. Even more im­port­ant is the lin­ear Schrödinger equa­tion, which we will nor­mal­ize here as $$\begin{array}{}\text{(4)}& i{\partial }_{t}u+\frac{1}{2}\mathrm{\Delta }u=0,\end{array}$$ where the un­known field $u:\mathbb{R}\times {\mathbb{R}}^3\to \mathbb{C}$ is now com­plex-val­ued. There are also non­lin­ear vari­ants of these equa­tions, such as the non­lin­ear Klein–Gor­don equa­tion $$\begin{array}{}\text{(5)}& -{\partial }_{tt}u+\mathrm{\Delta }u=m^{2}u+\lambda |u{|}^{p-1}u,\end{array}$$ and the non­lin­ear Schrödinger equa­tion $$\begin{array}{}\text{(6)}& i{\partial }_{t}u+\frac{1}{2}\mathrm{\Delta }u=\lambda |u{|}^{p-1}u,\end{array}$$ where $\lambda =\pm 1$ and $p>1$ are spe­cified para­met­ers. There are count­less oth­er fur­ther vari­ations (both lin­ear and non­lin­ear) of these dis­pers­ive equa­tions, such as Ein­stein’s equa­tions of gen­er­al re­lativ­ity, or the Korteweg–de Vries equa­tions for shal­low wa­ter waves.

An im­port­ant way to cap­ture dis­per­sion math­em­at­ic­ally is through the es­tab­lish­ment of dis­pers­ive in­equal­it­ies that as­sert, roughly speak­ing, that if a solu­tion $u$ to one of these equa­tions is suf­fi­ciently loc­al­ized in space at an ini­tial time, $t=0$, then it will de­cay as $t\to \mathrm{\infty }$. (If a solu­tion $u$ is not loc­al­ized enough in space ini­tially, it does not need to de­cay; con­sider for in­stance the trav­el­ing wave solu­tion $$u(t,x)=F(t-x_1)$$ to the wave equa­tion $\text{(1)}$.) This de­cay has to be meas­ured in suit­able func­tion space norms, such as the $L_x^{\mathrm{\infty }}({\mathbb{R}}^3)$ norm.

One can rep­res­ent any solu­tion $u$ to the lin­ear Schrödinger equa­tion ex­pli­citly in terms of the ini­tial data $u(0)$ by the for­mula $$u(t,x)=\frac{1}{(2\pi it{)}^{3/2}}{\int }_{{\mathbb{R}}^3}{e}^{-i|x-y{|}^2/2t}u(0,y)dy$$ for all $t\ne 0$ and $x\in {\mathbb{R}}^3$, where the quant­ity $(2\pi it{)}^{3/2}$ is defined us­ing a suit­able branch cut. From the tri­angle in­equal­ity, this im­me­di­ately gives the dis­pers­ive in­equal­ity $$\begin{array}{}\text{(7)}& \parallel u(t){\parallel }_{L_x^{\mathrm{\infty }}({\mathbb{R}}^3)}\le \frac{1}{(2\pi |t|{)}^{3/2}}\parallel u(0){\parallel }_{L_x^1({\mathbb{R}}^3)}.\end{array}$$ If the solu­tion is ini­tially spa­tially loc­al­ized in the sense that the $L^1$ norm $$\parallel u(0){\parallel }_{L_x^1({\mathbb{R}}^3)}$$ is fi­nite, then the solu­tion $u(t)$ de­cays uni­formly to zero as $t\to \pm \mathrm{\infty }$. A sim­il­ar (but slightly more com­plic­ated) dis­pers­ive in­equal­ity can also be ob­tained for solu­tions to the lin­ear Klein–Gor­don equa­tion $\text{(3)}$.

On the oth­er hand, solu­tions to the lin­ear Schrödinger equa­tion $\text{(4)}$ sat­is­fy the point­wise mass con­ser­va­tion law $$\begin{array}{}\text{(8)}& {\partial }_t|u{|}^2=\sum _{j=1}^3{\partial }_{x_j}\mathrm{Im}(\overline{u}{\partial }_{x_j}u).\end{array}$$ From this, one can eas­ily de­rive con­ser­va­tion of the spa­tial $L^2$ norm of the solu­tion: $$\parallel u(t){\parallel }_{L_x^2({\mathbb{R}}^3)}=\parallel u(0){\parallel }_{L_x^2({\mathbb{R}}^3)}.$$ In par­tic­u­lar, the $L^2$ norm of the solu­tion will stay con­stant in time, rather than de­cay to zero.

To re­con­cile this fact with the dis­pers­ive es­tim­ate, we ob­serve that solu­tions to dis­pers­ive equa­tions such as the lin­ear Schrödinger equa­tion spread out in space as time goes to in­fin­ity (much as the ripples on a pond do), al­low­ing the $L^{\mathrm{\infty }}$ norm of such a solu­tion to go to zero even while the $L^2$ norm stays bounded away from zero. As men­tioned earli­er, this ef­fect can also be seen for the wave equa­tion $\text{(1)}$.

The above ana­lys­is of the lin­ear Schrödinger equa­tion re­lied cru­cially on hav­ing an ex­pli­cit fun­da­ment­al solu­tion at hand. What hap­pens if one works with non­lin­ear (and not com­pletely in­teg­rable) equa­tions, such as $\text{(5)}$ or $\text{(6)}$, in which no ex­pli­cit and tract­able for­mula for the solu­tion is avail­able? For lin­ear equa­tions (such as the wave or Schrödinger equa­tion out­side of an obstacle, or in the pres­ence of po­ten­tials or mag­net­ic fields) one can still hope to use meth­ods from spec­tral the­ory to un­der­stand the long-time be­ha­vi­or (as is done for in­stance in the fam­ous RAGE the­or­em of Ruelle (1969), Am­rein–Georges­cu (1973), and Enss (1977)). However, such meth­ods are ab­sent for non­lin­ear equa­tions such as $\text{(5)}$ or $\text{(6)}$, par­tic­u­larly when deal­ing with solu­tions that are too large for per­turb­at­ive the­ory to be of much use.

Re­call that in 1961, Mor­awetz proved the de­cay of solu­tions to the clas­sic­al wave equa­tion in the pres­ence of a star-shaped obstacle. Mor­awetz used the “Friedrichs abc meth­od,” in which one mul­ti­plied both sides of a PDE such as $\text{(5)}$ or $\text{(6)}$ by a mul­ti­pli­er $$a{\partial }_{t}u+b\cdot \mathrm{\nabla }u+cu$$ for well-chosen func­tions $a$, $b$, $c$, in­teg­rated over a space-time do­main, and re­arran­ging us­ing in­teg­ra­tion by parts and omit­ting some terms of def­in­ite sign, ob­tained a use­ful in­teg­ral in­equal­ity. The key dis­cov­ery of Mor­awetz (a ver­sion of which first ap­peared in work of Lud­wig) was that this meth­od was par­tic­u­larly fruit­ful when the mul­ti­pli­er was equal to the ra­di­al de­riv­at­ive $$\frac{x\cdot \mathrm{\nabla }u}{|x|}$$ of the solu­tion (in some cases one also adds a lower-or­der term $u/|x|$).

In 1968, Mor­awetz ap­plied this tech­nique to study solu­tions $u$ to the non­lin­ear Klein–Gor­don equa­tion $\text{(5)}$, as­sum­ing one is in the non­focus­ing case with $\lambda =m=+1$. (In the fo­cus­ing case $\lambda =-1$, the equa­tion $\text{(5)}$ ad­mits “soliton” solu­tions that are sta­tion­ary in time and thus do not dis­perse.) By us­ing a mul­ti­pli­er of the above form, Mor­awetz ob­tained an in­equal­ity of the form $$\begin{array}{}\text{(9)}& {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^3}U(t,x)dxdt\le CE(u(0)),\end{array}$$ where $$U(t,x)=\frac{|u(t,x){|}^2+|u(t,x){|}^{p+1}}{|x|}.$$ Here the con­stant $C$ de­pends only on the ex­po­nent $p$ and $E(u(0))$ is the en­ergy: $$E(u(0))={\int }_{{\mathbb{R}}^3}\frac{1}{2}|\mathrm{\nabla }u(0,x){|}^2+\frac{1}{2}|{\partial }_{t}u(0,x){|}^2+\frac{1}{p+1}|u(0,x){|}^{p+1}dx.$$ This type of es­tim­ate is now known as a Mor­awetz in­equal­ity. The key point here is that the left-hand side of the Mor­awetz in­equal­ity in $\text{(9)}$ con­tains an in­teg­ra­tion over the en­tire time do­main $\mathbb{R}$ (as op­posed to a time in­teg­ral over a bounded in­ter­val). It im­me­di­ately rules out soliton-type solu­tions that move at bounded speed (as this would make the left-hand side of $\text{(9)}$ in­fin­ite). It forces some time-av­er­aged de­cay of the solu­tion near the spa­tial ori­gin $x=0$. For in­stance, it is im­me­di­ate from $\text{(9)}$ that $$\frac{1}{T}{\int }_0^T{\int }_K|u(t,x){|}^{2}dxdt\to 0$$ as $T\to \mathrm{\infty }$ for any com­pact spa­tial re­gion $K\subset {\mathbb{R}}^3$.

Once one has some sort of de­cay es­tim­ate for a dis­pers­ive equa­tion, it is of­ten pos­sible to “boot­strap” the es­tim­ate to ob­tain ad­di­tion­al de­cay es­tim­ates. For in­stance one might use the de­cay es­tim­ate one already has to bound the right-hand side of a non­lin­ear PDE such as $\text{(5)}$ or $\text{(6)}$, and then solve the as­so­ci­ated (in­homo­gen­eous) lin­ear PDE to ob­tain a new de­cay es­tim­ate for the solu­tion.

An early res­ult of this type was de­veloped by Mor­awetz and Strauss in 1975. They showed that for any fi­nite en­ergy solu­tion $u$ to the non­lin­ear Klein–Gor­don equa­tion $\text{(5)}$ with $\lambda =m=+1$ and $p=3$, the solu­tion de­cays like a solu­tion to the lin­ear Klein–Gor­don equa­tion $\text{(3)}$. More pre­cisely, there ex­ist fi­nite solu­tions $u_{+},u_{-}$ to $\text{(3)}$ such that $u(t)-u_{+}(t)$ (resp. $u(t)-u_{-}(t)$) goes to zero in the en­ergy norm as $t\to +\mathrm{\infty }$ (resp. $t\to -\mathrm{\infty }$). This can be de­veloped fur­ther in­to a sat­is­fact­ory scat­ter­ing the­ory for such equa­tions, which among oth­er things gives a con­tinu­ous scat­ter­ing map from $u_{-}$ to $u_{+}$ or vice versa. See Fig­ure 2.

In the dec­ades since Mor­awetz’s pi­on­eer­ing work, many ad­di­tion­al Mor­awetz in­equal­it­ies have been de­veloped. For in­stance, in 1978, Lin and Strauss de­veloped Mor­awetz in­equal­it­ies for the non­lin­ear Schrödinger equa­tion, and Mor­awetz her­self dis­covered fur­ther such es­tim­ates for the wave equa­tion out­side of an obstacle. In more re­cent years, “in­ter­ac­tion Mor­awetz in­equal­it­ies” were in­tro­duced, which could con­trol cor­rel­a­tion quant­it­ies such as $$\begin{array}{}\text{(10)}& {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\frac{|u(t,x){|}^2|u(t,y){|}^p}{|x-y|}dxdydt\end{array}$$ for solu­tions $u$ to the non­lin­ear Schrödinger equa­tion $\text{(6)}$.

One way to view Mor­awetz in­equal­it­ies is as an as­ser­tion of mono­ton­icity of the ra­di­al mo­mentum, which takes the form $${\int }_{{\mathbb{R}}^3}({\partial }_{t}u)(\frac{x}{|x|}\cdot \mathrm{\nabla }u)dx$$ for wave or Klein–Gor­don equa­tions, and $${\int }_{{\mathbb{R}}^3}\mathrm{Im}(\overline{u}\frac{x}{|x|}\cdot \mathrm{\nabla }u)dx$$ for Schrödinger equa­tions. In­form­ally, this quant­ity is ex­pec­ted to be pos­it­ive when waves propag­ate away from the ori­gin, and neg­at­ive when they propag­ate to­wards the ori­gin. The in­tu­ition is that while waves can some­times propag­ate to­wards the ori­gin, even­tu­ally they will move past the ori­gin and be­gin ra­di­at­ing away from the ori­gin. However, in the ab­sence of fo­cus­ing mech­an­isms (such as a neg­at­ive sign $\lambda =-1$ in the non­lin­ear­ity), the re­verse phe­nomen­on of out­ward net ra­di­al mo­mentum be­ing con­ver­ted to in­ward net ra­di­al mo­mentum can­not oc­cur. Thus the ra­di­al mo­mentum is al­ways ex­pec­ted to be in­creas­ing in time.

On the oth­er hand, un­der hy­po­theses such as fi­nite en­ergy, this ra­di­al mo­mentum should be bounded. So by the fun­da­ment­al the­or­em of cal­cu­lus, the time de­riv­at­ive of the ra­di­al mo­mentum should have a bounded in­teg­ral in time. In­tu­it­ively, one ex­pects this time de­riv­at­ive to be large when the solu­tion has a strong pres­ence near the ori­gin, but not when the solu­tion is far away from the ori­gin. Far from the ori­gin the ra­di­al vec­tor field $$\frac{x}{|x|}\cdot \mathrm{\nabla }$$ be­haves like a con­stant, and the ra­di­al mo­mentum ap­proaches a fixed co­ordin­ate of the total mo­mentum. This ex­plains why Mor­awetz in­equal­it­ies tend to in­volve factors such as $1/|x|$ that loc­al­ize the es­tim­ate to near the ori­gin.

The Mor­awetz in­equal­it­ies are in­dis­pens­ible as an in­gredi­ent in con­trolling the long-time be­ha­vi­or of solu­tions to a wide ar­ray of dis­pers­ive de­fo­cus­ing equa­tions, in­clud­ing a num­ber of en­ergy-crit­ic­al or mass-crit­ic­al equa­tions in which the ana­lys­is is par­tic­u­larly del­ic­ate and in­ter­est­ing; see for in­stance the texts [e1], [e3], [e4] for de­tailed cov­er­age of these top­ics. They have also been suc­cess­fully ap­plied to many equa­tions in gen­er­al re­lativ­ity (such as Ein­stein’s equa­tions for grav­it­a­tion­al fields), for in­stance to ana­lyze the asymp­tot­ic be­ha­viour around a black hole. The fun­da­ment­al tools that Mor­awetz has in­tro­duced to the field of dis­pers­ive equa­tions will cer­tainly un­der­lie fu­ture pro­gress in this field for dec­ades to come.