Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 (Introduction to Nonlinear Analysis)

Write the Generalized Korteweg–De Vries equation asFor a sufficiently regular solution, integration by parts givesThus KdV mass conservation givesis conserved.

Differentiate the proposed energy and integrate the kinetic term by parts:Putting $F=u_{xx}+u^p$, the equation says $u_t=-F_x$, soHence KdV energy conservation gives

The one-dimensional Gagliardo-Nirenberg interpolation inequality yieldsMass conservation fixes $\Vert u{\Vert }_2$. If $X=\Vert u_x{\Vert }_2$, energy conservation therefore givesFor $p<5$, the second exponent is strictly smaller than two. The right side tends to infinity with $X$, so this inequality bounds $X$ uniformly throughout the lifespan. The conserved $L^2$ norm then bounds $\Vert u(t){\Vert }_{H^1}$. The stated blowup criterion rules out a finite endpoint, proving Global existence for the energy-subcritical generalized KdV equation.

Set $z=x-ct$ and $u(t,x)=Q_c(z)$. Substitution into the equation givesDecay at infinity makes the integration constant zero. Ifthen every term in $Q_c^{\prime \prime }-c{Q}_c+Q_c^p$ equals $c^{1+1/(p-1)}$ times the corresponding term in $Q^{\prime \prime }-Q+Q^p$. Thus the required Generalized KdV solitary wave is

For every $p<5$, the wave $Q_c$ has Orbital stability of a generalized KdV solitary wave in $H^1(\mathbb{R})$ modulo translation: for every $\epsilon >0$ there is $\delta >0$ such thatThe relevant stability slope has the correct sign because scaling giveswhich is strictly increasing in $c$ precisely for $p<5$. Together with the constrained variational characterization of $Q_c$, the conserved mass and energy provide a coercive Lyapunov function transverse to the translation direction.

The Gravitational Hartree equation can be writtenwhere $\varphi$ is real. Consequentlybecause both integrals multiplied by $i$ are purely imaginary after integration by parts. Hence Hartree mass conservation holds.

Since $\Delta \varphi =|u{|}^2$, integration by parts givesThe Newtonian convolution operator is self-adjoint, so differentiating this identity symmetrically givesIt follows thatThe equation says $-\Delta u+\varphi u=i{u}_t$, and therefore the final real part is $\mathrm{Re}\int i|u_t{|}^2=0$. This proves Hartree energy conservation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality applied to the Newtonian kernel givesThe Gagliardo-Nirenberg interpolation inequality then givesWriting $X=\Vert \nabla u{\Vert }_2$ and using the conserved mass, conservation of energy impliesThus $X$ remains bounded. Together with the conserved $L^2$ norm this bounds $\Vert u(t){\Vert }_{H^1}$, and the supplied blowup criterion proves Global H1 solutions of the three-dimensional gravitational Hartree equation.

LetThe first virial identity, obtained from the equation by integration by parts, isDifferentiating once more givesWrite $\varphi =W\ast |u{|}^2$, where $W(x)=-1/(C_4|x{|}^2)$ is homogeneous of degree $-2$. Symmetrizing the double integral and applying Euler's identity $z\cdot \nabla W(z)=-2W(z)$ yieldsTherefore

Not all solutions are global. Choose smooth finite-variance data of negative energy, which is possible by multiplying any nonzero test function by a sufficiently large constant: the kinetic term is quadratic in the amplitude and the attractive potential term is quartic. If such a solution were global, the Virial identity for the four-dimensional gravitational Hartree equation would make the nonnegative function $I(t)$ strictly concave with constant negative second derivative, forcing it below zero in finite time. The solution must therefore blow up in finite time.

Letbe the harmonic-oscillator energy space, and denote the minimized quadratic functional by $J(u)$. A minimizing sequence $(u_n)\subset \mathcal{A}(M)$ is bounded in $\Sigma$, because $J$ is its squared Hilbert norm with positive coefficients. After taking a subsequence, $u_n\rightharpoonup u$ weakly in $\Sigma$.

The embedding $\Sigma \hookrightarrow L^4({\mathbb{R}}^2)$ is compact. On any fixed ball this follows from the Rellich-Kondrachov compactness theorem. Outside a large ball, the moment bound makes the $L^2$ tail uniformly small, and interpolation with the uniform $H^1\hookrightarrow L^q$ bounds for some $q>4$ makes the $L^4$ tail uniformly small. Hence $u_n\to u$ strongly in $L^4$.

The constraint passes to the limit, so $\int |u{|}^4=M$. Weak lower semicontinuity gives $J(u)\le \mathrm{liminf}J(u_n)=I(M)$. Therefore $u$ attains the infimum. This is Fixed-L4 minimization in the harmonic-oscillator energy space.

Replacing a minimizer by its absolute value does not increase its gradient norm, so choose a nonnegative minimizer $P$. The Euler-Lagrange equation isfor a Lagrange multiplier $\mu$. Multiplication by $P$ and integration show thatso $\mu >0$. Set $P_{\eta }=\sqrt{\mu }P$. Then $P_{\eta }$ is nonzero, belongs to $H^1({\mathbb{R}}^2)$, and satisfies the trapped focusing cubic ground-state equationStandard elliptic regularity and the strong minimum principle for elliptic operators make the nonnegative solution positive.

From $b_s+b^2=-\eta$ one obtainsSince ${\lambda }_s=-b\lambda$, the ratio $\sqrt{b^2+\eta }/\lambda$ is constant. Its value at $t=-1$ is one, soUsing $s_t={\lambda }^{-2}$ and ${\lambda }_s=-b\lambda$ givesThe initial conditions now giveFinally,These are the explicit lens parameters.

Substitute the Focusing Schrödinger lens ansatz intoThe imaginary terms proportional to $P_{\eta }+y\cdot \nabla P_{\eta }$ cancel because $b=-\lambda {\lambda }_t$. After multiplying the remaining real equation by ${\lambda }^3$, one obtainsThe dynamical system gives ${\lambda }^2{b}_t+b^2=-\eta$. Comparison with the equation for $P_{\eta }$ therefore requiresThus, up to an arbitrary constant phase,

For $T_2>T_1\ge -1$, the dual Strichartz estimate for the free Schrödinger equation givesThe assumed finite global $L_{t,x}^4$ norm makes the right side tend to zero as $T_1,T_2\to \infty$. The displayed family is therefore a Cauchy sequence in the complete space $L^2$ and has a strong limit $F_{\infty }$.

The Duhamel principle givesDefineSince the free Schrödinger group is unitary,This is scattering from a finite Strichartz norm.