arXiv subjects

Enno Lenzmann

Publications and source records attributed to Enno Lenzmann.

At least 19 recordsLinked to original sources

arXiv2026

Finite-time blow-up for the Calogero--Sutherland derivative NLS on $\mathbb{T}$

We study finite-time blow-up for the focusing Calogero--Sutherland derivative NLS on the torus with smooth initial data in the Hardy space $L^2_+(\mathbb{T})$. For finite-gap potentials as initial data, the explicit solution formula reduces the dynamics to a finite-dimensional analytic family of contractions. This yields a complete description of every finite-time blow-up solution in this class: near blow-up time $T >0$, the solution decomposes into finitely many concentrating universal blow-up profiles and a smooth remainder, each bubble carries one unit of $L^2$-mass, the concentration points are distinct, and the blow-up rates are quantized according to $$ \|u(t) \|_{H^s(\mathbb{T})} \sim_{s,u_0} \frac{1}{(T-t)^{2s\nu}} \quad \mbox{as} \quad t \nearrow T \quad \mbox{for any $s > 0$} $$ with some integer $\nu\geq1$. Within an explicit subclass of finite-gap potentials, we identify an exact resonance condition that yields the existence of finite-time blow-up solutions. By a non-perturbative method, we construct single- and multi-bubble blow-up solutions with rate $\nu=1$ for every prescribed $L^2$-mass in $(1,\infty)\setminus \{ 2 \}$. In the complementary non-resonant regime, we prove global existence with uniform Sobolev bounds. The construction also shows instability of the blow-up solutions and non-chiral finite-time blow-up examples. To the best of our knowledge, these results provide the first explicit, non-perturbative construction of finite-time blow-up for an integrable NLS-type equation on the torus, together with a complete classification of the singular dynamics within a natural finite-gap class.

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arXiv2026

The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

We consider the half-wave maps equation $$ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} $$ for $\mathbf{u} : \mathbb{R} \times \mathbb{T} \to \mathbb{S}^2$, where $\mathbb{T}=\mathbb{R}/2 \pi \mathbb{Z}$ is the one-dimensional torus and $\mathbb{S}^2 \subset \mathbb{R}^3$ denotes the unit sphere. By extension from rational initial data, we construct a unique and continuous flow map for data in the critical energy space $H^{1/2}(\mathbb{T}; \mathbb{S}^2)$. Moreover, we show almost periodicity in time of these solutions. For the dense subset of rational initial data, we establish quasi-periodicity in time and a-priori bounds on $\| \mathbf{u}(t) \|_{H^s(\mathbb{T})}$ for any $s >0$. Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of vector-valued holomorphic functions on the unit disk. As a central ingredient, we develop a general {\em stability principle} for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin--Ono equation, Calogero--Sutherland DNLS, and the half-wave-maps equation posed on $\mathbb{T}$. Our results extend to the matrix-valued half-wave maps equation $$ \partial_t \mathbf{U} = -\frac{i}{2} [ \mathbf{U}, |D| \mathbf{U} ] $$ with target manifold given by the complex Grassmannians $\mathsf{Gr}_k(\mathbb{C}^d)$, thereby generalizing the special case $\mathbb{S}^2 \cong \mathbb{CP}^1 \cong \mathsf{Gr}_1(\mathbb{C}^2)$. In a companion work, we prove global well-posedness for the half-wave maps equation posed on $\mathbb{R}$ in the scaling-critical energy space $\dot{H}^{1/2}$, by establishing a stability principle for explicit formulae on Hardy spaces in the complex half-plane $\mathbb{C}_+$.

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arXiv2025

Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$

We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation $$ (-\Delta)^s u = e^u \quad \mbox{in $\mathbb{R}$} \quad \mbox{with} \quad \int_{\mathbb{R}} e^u dx < +\infty $$ for all exponents $s \in (\frac{1}{2},1)$. Furthermore, we show $u$ has finite Morse index and that its linearized operator is nondegenerate. Our arguments are based on a fixed point scheme in terms of the function $v= \sqrt{e^u}$ and we devise a nonlocal shooting method involving (locally) compact nonlinear maps. We also study existence, symmetry and uniqueness of solutions to $(-\Delta)^s u = K e^u$ in $\mathbb{R}$ with $K e^u \in L^1(\mathbb{R})$ for a general class of positive, even and monotone-decreasing functions $K > 0$.

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arXiv2024

Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf{u}_0 : \mathbb{R} \to \mathbb{S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $\dot{H}^{1/2}(\mathbb{R}; \mathbb{S}^2)$. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms $\dot{H}^s$ for $s > 0$. Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $\mathbf{Gr}_k(\mathbb{C}^d)$. Notably, this includes the complex projective spaces $ \mathbb{CP}^{d-1} \cong \mathbf{Gr}_1(\mathbb{C}^d)$ thereby extending the classical case of the target $\mathbb{S}^2 \cong \mathbb{CP}^1$.

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arXiv2022

The Calogero--Moser Derivative Nonlinear Schr\"odinger Equation

We study the Calogero--Moser derivative NLS equation $$ i \partial_t u +\partial_{xx} u + (D+|D|)(|u|^2) u =0 $$ posed on the Hardy-Sobolev space $H^s_+(\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical equation, we prove global well-posedness for $s \geq 1$ and initial data with sub-critical or critical $L^2$-mass $\| u_0 \|_{L^2}^2 \leq 2 \pi$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we prove that they exhibit energy cascades in the following strong sense such that $\|u(t)\|_{H^s} \sim_s |t|^{2s}$ as $t \to \pm \infty$ for every $s > 0$. \end{abstract}

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arXiv2022

Uniqueness for the Nonlocal Liouville Equation in $\mathbb{R}$

We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-Δ)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total $Q$-curvature $\int_{\mathbb{R}} K e^w \, dx< +\infty$. Here the prescribed $Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with $K(x) = \exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero--Moser derivative NLS, which is a completely integrable PDE recently studied by P. Gérard and the second author.

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arXiv2022

On symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.\,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szegö inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $\mathbb{R}^n$ recently obtained by the last two authors of the present paper.

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arXiv2021

Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates

We consider ground states solutions $u \in H^2(\mathbb{R}^N)$ of biharmonic (fourth-order) nonlinear Schrödinger equations of the form $$ Δ^2 u + 2a Δu + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with positive constants $a, b > 0$ and exponents $2 < p < 2^*$, where $2^* = \frac{2N}{N-4}$ if $N > 4$ and $2^* = \infty$ if $N \leq 4$. By exploiting a connection to the adjoint Stein--Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states $u\in H^2(\mathbb{R}^N)$ in dimension $N \geq 2$ fail to be radially symmetric for all exponents $2 < p < \frac{2N+2}{N-1}$ in a suitable regime of $a,b>0$. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained $L^2$-mass and for a related problem on the unit ball in $\mathbb{R}^N$ subject to Dirichlet boundary conditions.

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arXiv2020

Derivation of the Half-Wave Maps Equation from Calogero--Moser Spin Systems

We prove that the energy-critical half-wave maps equation \[ \partial_t \mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R} \times \mathbb{T} \] arises as an effective equation in the continuum limit of completely integrable Calogero-Moser classical spin systems with inverse square $1/r^2$ interactions on the circle. We study both the convergence to global-in-time weak solutions in the energy class as well as short-time strong solutions of higher regularity. The proofs are based on Fourier methods and suitable discrete analogues of fractional Leibniz rules and Kato-Ponce-Vega commutator estimates. In a companion paper, we further extend our arguments to study the real line case and more general spin interactions.

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arXiv2019

On symmetry of traveling solitary waves for dispersion generalized NLS

We consider dispersion generalized nonlinear Schrödinger equations (NLS) of the form $i \partial_t u = P(D) u - |u|^{2 σ} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $σ\in \mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.

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arXiv2019

A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order

We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of arbitrary order, ii) ground states for bi- and polyharmonic NLS, and iii) Adams-Moser-Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geq 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy-Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.

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arXiv2019

A Short Primer on the Half-Wave Maps Equation

We review the current state of results about the half-wave maps equation on the domain $\mathbb{R}^d$ with target $\mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves and a Lax pair structure together with its implications (e.\,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.

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arXiv2018

Existence and stability of Schrödinger solitons on noncompact manifolds

We consider the focusing nonlinear Schrödinger equation on a large class of rotationally symmetric, noncompact manifolds. We prove the existence of a solitary wave by perturbing off the flat Euclidean case. Furthermore, we study the stability of the solitary wave under radial perturbations by analyzing spectral properties of the associated linearized operator. Finally, in the L2-critical case, by considering the Vakhitov-Kolokolov criterion (see also results of Grillakis-Shatah-Strauss), we provide numerical evidence showing that the introduction of a nontrivial geometry destabilizes the solitary wave in a wide variety of cases, regardless of the curvature of the manifold. In particular, the parameters of the metric corresponding to standard hyperbolic space will lead to instability consistent with the blow-up results of Banica-Duyckaerts (2015). We also provide numerical evidence for geometries under which it would be possible for the Vakhitov-Kolokolov condition to suggest stability, provided certain spectral properties hold in these spaces

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arXiv2018

On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-Δ} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iωt} Q_v(x-vt) $$ with frequency $ω\in \R$, velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$, $Q_v \not \equiv 0$. We prove that traveling solitary waves for speeds $|v| \geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $\sqrt{-\DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d \geq 2$.

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arXiv2017

A Lax Pair Structure for the Half-Wave Maps Equation

We consider the half-wave maps equation $$ \partial_t \vec{S} = \vec{S} \wedge |\nabla| \vec{S}, $$ where $\vec{S}= \vec{S}(t,x)$ takes values on the two-dimensional unit sphere $\mathbb{S}^2$ and $x \in \mathbb{R}$ (real line case) or $x \in \mathbb{T}$ (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in \cite{LS,Zh}, which formally arises as an effective evolution equation in the classical and continuum limit of Haldane-Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target $\mathbb{H}^2$ (hyperbolic plane).

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arXiv2017

On energy-critical half-wave maps into $\mathbb{S}^2$

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $\mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Möbius group on the target $\mathbb{S}^2$ applied to half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $\mathbf{Q}$ with arbitrary degree $m \geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $\mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $\mathbb{R}$ to the unit circle $\mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.

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arXiv2016

Sharp commutator estimates via harmonic extensions

We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da Lio-Rivière three-term commutators. We also give a new limiting $L^1$-estimate for a double commutator of Coifman-Rochberg-Weiss-type, and several intermediate estimates. The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general paraproduct arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.

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arXiv2016

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line

We consider the focusing cubic half-wave equation on the real line $$i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(ξ)=|ξ|\hat{u}(ξ), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.

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