Nonlinear analysis

Nonlinear analysis studies equations and variational problems in which superposition fails. Its tools include nonlinear functional inequalities, conserved energies, compactness, phase-plane methods, and concentration arguments.

For $0<\lambda <d$ and suitable conjugate exponents, the Hardy–Littlewood–Sobolev inequality bounds convolution with the Riesz kernel $|x{|}^{-\lambda }$. One useful form is

The generalized Korteweg–De Vries equation is the nonlinear partial differential equationThe cases below the exponent $p=5$ are energy-subcritical in $H^1(\mathbb{R})$.

Sufficiently regular decaying solutions conserveBoth identities follow by integration by parts from the divergence form of the equation.

For $p<5$, the one-dimensional Gagliardo-Nirenberg interpolation inequality givesThe exponent of $\Vert u_x{\Vert }_2$ is below two, so the conserved mass and energy control the $H^1$ norm. The blowup alternative then extends every local $H^1$ solution globally.

If $Q^{\prime \prime }-Q+Q^p=0$, thensatisfies $Q_c^{\prime \prime }-c{Q}_c+Q_c^p=0$, and $u(t,x)=Q_c(x-ct)$ is a traveling-wave solution of the Generalized Korteweg–De Vries equation.

For $p<5$, a generalized KdV solitary wave is orbitally stable in $H^1(\mathbb{R})$ modulo spatial translations. The variational slope has the stable sign becauseis strictly increasing in $c$. Conserved mass and energy then furnish a Lyapunov functional transverse to the translation orbit.

This attractive mean-field equation is a gravitational version of the Hartree equations. The gravitational Hartree equation couples a wave function to its attractive Newtonian potential:

The gravitational Hartree flow conservesThe self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\mathrm{Re}\int \varphi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality and Gagliardo-Nirenberg interpolation inequality giveConserved mass and energy therefore bound the $H^1$ norm by Young inequality, and the blowup alternative gives global existence.

For a finite-variance solution in four dimensions,satisfies $I^{\prime \prime }(t)=16E(u)$. The identity uses the degree $-2$ homogeneity of the Newtonian kernel. Negative-energy finite-variance data consequently blow up in finite time by the convexity argument.

The Emden-Fowler transformation is a change of variables for scaling-invariant radial equations, including the Lane-Emden equation. The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

The classical Lane-Emden equation is the dimensionless equationfor a spherically symmetric polytrope in three dimensions. The Generalized Lane-Emden equation extends it to arbitrary dimensions and to functions without radial symmetry.

A Generalized Lane-Emden equation is a semilinear partial differential equation whose highest-order term is the Laplacian:For a radial function in $d$ dimensions it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$. The case $d=3$ gives the classical Lane-Emden equation.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.

A Pohozaev identity is an integral identity for a nonlinear partial differential equation, obtained by multiplying the equation by a scaling vector field such as $x\cdot \nabla u$ and applying integration by parts. It relates kinetic, potential, and nonlinear energies.

Testing the ground-state equation against $Q$ and applying the Pohozaev identity givesConsequently $E(Q)=0$ and $\inf J=\frac{d}{d+2}\Vert Q{\Vert }_2^{4/d}$.

The symmetric decreasing rearrangement $u^{\ast }$ of a measurable function is the radial, radially nonincreasing function whose superlevel sets are centered balls with the same measures as those of $|u|$. It preserves $L^p$ norms, and the Pólya-Szegő inequality gives $\Vert \nabla u^{\ast }{\Vert }_2\le \Vert \nabla u{\Vert }_2$.

For $u\in W^{1,p}({\mathbb{R}}^d)$, the symmetric decreasing rearrangement satisfies

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

The mass-critical focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$ isIts scaling $u(t,x)\mapsto {\lambda }^{d/2}u({\lambda }^{2}t,\lambda x)$ preserves the Lebesgue space norm $\Vert u{\Vert }_{L^2}$, which is the conserved mass.

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

A virial identity describes the time derivatives of a spatial moment of a solution to a dispersive equation. For the mass-critical nonlinear Schrödinger equation, the second derivative of the variance is a constant multiple of the conserved energy.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

For a finite-variance solution of the mass-critical focusing nonlinear Schrödinger equation, the virial identity givesIf $E(u_0)<0$, the right-hand side is a negative constant. The nonnegative variance would then become negative in finite time if the solution remained regular, so the solution blows up in finite time.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For $p=2+4/d$, the mass-critical Weinstein functional isIt is invariant under multiplication by a nonzero scalar, spatial dilation, translation, and constant phase. Its positive minimizers are rescalings and translations of the NLS ground state.

Normalize a minimizing sequence so that its $L^2$ norm and gradient norm are fixed. Symmetric decreasing rearrangement does not increase the gradient norm and preserves every $L^p$ norm, while radial compactness prevents translation loss. A weakly convergent subsequence therefore converges strongly in the nonlinear $L^{2+4/d}$ norm and yields a nonnegative minimizer.

Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

In dimension $d$, the mass-critical sharp inequality isand equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

The sharp inequality impliesThus energy controls the gradient whenever the mass is strictly below the mass of the NLS ground state.

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.

Every bounded sequence $(u_n)$ in $H^1({\mathbb{R}}^d)$ has, after passage to a subsequence, a decompositionwhere $|x_n^j-x_n^k|\to \infty$ for $j\ne k$, the $L^2$ and gradient norms decouple asymptotically, andfor every $2<p<2^{\ast }$.

This local phase change is a gauge transformation of the wave function. Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

The trapped focusing cubic ground-state equation in two dimensions isA nonzero positive solution follows from minimizing the harmonic-oscillator quadratic form subject to a fixed $L^4$ norm and then rescaling by the positive Lagrange multiplier.

If $P_{\eta }$ solves the trapped focusing cubic ground-state equation, the ansatzsolves the two-dimensional mass-critical focusing nonlinear Schrödinger equation when

The initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ selectThe scale grows linearly at infinity, yielding a finite critical spacetime norm for the corresponding lens solution.

Young's inequality for products states that for conjugate exponents $q,q^{\prime }>1$ and nonnegative $a,b$,Its weighted form absorbs a lower power into a coercive higher power at the cost of a constant.

For a nonnegative potential $V(x)\to \infty$ as $|x|\to \infty$, the confining-potential energy space isA local Poincaré inequality controls the missing $L^2$ term, so this energy norm makes ${\Sigma }_V$ a Hilbert space.

The embedding ${\Sigma }_V\hookrightarrow L^2({\mathbb{R}}^d)$ is compact. The Rellich-Kondrachov compactness theorem gives compactness on each fixed ball, whilemakes the tails uniformly small.

In two dimensions, the quadratic functionalattains its minimum subject to $\Vert u{\Vert }_4^4=M>0$. A bounded minimizing sequence is weakly compact in the harmonic-oscillator energy space and strongly compact in $L^4$: local Rellich-Kondrachov compactness theorem and the weighted tail bound prevent escape to infinity.

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.

The two-dimensional defocusing cubic nonlinear Schrödinger equation is mass-critical: the scaling $u(t,x)\mapsto {\lambda }^{-1}u(t/{\lambda }^2,x/\lambda )$ preserves the $L^2$ norm. The defocusing sign makes its nonlinear energy positive.

The harmonic-oscillator energy space isThe spatial moment prevents mass from escaping to infinity.

The embedding $\Sigma \hookrightarrow L^2({\mathbb{R}}^d)$ is compact. Rellich compactness controls a fixed ball, whilecontrols the tail uniformly.

Critical points of the trapped defocusing cubic energysatisfy $-\Delta v+|x{|}^{2}v-\omega v+v^3=0$.

If $v$ solves the trapped stationary equation, the rescaled quadratic-phase ansatzsolves the free mass-critical equation when the scaling, chirp, and phase parameters satisfy the associated lens-transform system.

In two dimensions, the endpoint spacetime estimate and its dual include

If a global cubic Schrödinger solution has finite $L_{t,x}^4$ norm on a time ray, the dual Strichartz estimate makes its interaction representation Cauchy in $L^2$. It therefore converges to scattering data $u_{\infty }$, and $u(t)-e^{it\Delta }{u}_{\infty }\to 0$ in $L^2$.