A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (1) the transport equation in phase space generalizing the classical Vlasov equation, (2) the linear Schrödinger equation, (3) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain \(L^1\)—\(L^\infty \) decay through directly estimating the fundamental solution in physical space or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines “vector field” commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space–time weighted energy estimates. Combined with a simple version of Sobolev’s inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrödinger equations, we can recover sharp pointwise decay; in the Schrödinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on \(L^2\)—\(L^\infty \) decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.     

Short-Time Nonlinear Effects in the Exciton-Polariton System

In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by \(\epsilon ^\alpha \), it is found that the relative error committed by omitting the nonlinear term in the exciton-polariton system remains within \(\epsilon \) for all times up to \(t=C\epsilon ^\beta \), where \(\beta =(1-\alpha (p-1))/(p+2)\). This is in contrast to \(\beta =1-\alpha (p-1)\) for the nonlinear Schrödinger equation. The result is proved for solutions in \(H^s(\mathbb {R}^n)\) for \(s>n/2\). Numerical computations indicate that the results are sharp and also hold in \(L^2(\mathbb {R}^n)\).     

A Spectral Multiplier Theorem Associated with a Schrödinger Operator

We establish a Hörmander type spectral multiplier theorem for a Schrödinger operator \(H=-\Delta +V(x)\) in \(\mathbb {R}^3\), provided V is contained in a large class of short range potentials. This result does not require the Gaussian heat kernel estimate for the semigroup \(e^{-tH}\), and indeed the operator H may have negative eigenvalues. As an application, we show local well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.     

Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity

In this paper, we investigate the following critical fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^su+V(x)u=|u|^{2_s^*-2}u+\lambda K(x)f(u), \ x \in \mathbb {R}^N, \end{aligned}$$

where \(\lambda >0\), \(0<s<1\), \((-\Delta )^s\) denotes the fractional Laplacian of order s, \(V, \ K\) are nonnegative continuous functions satisfying some conditions and f is a continuous function, \(N>2s\) and \(2_s^*=\frac{2N}{N-2s}\). We prove that the equation has a positive solution for large \(\lambda \) by the variational method.

     

Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in \(H^r\) for solutions in \(H^{r+1}\) (with \(r > d/2\)) of the derived schemes. This allows us lower regularity assumptions on the data than for instance required for classical splitting or exponential integration schemes. For one-dimensional quadratic Schrödinger equations, we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes.     

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion \({\textstyle \int }h_k\), \(k\in {\mathbb {Z}}_{+}\). In each \({\textstyle \int }h_{2k}\) the term with the highest regularity involves the Sobolev norm \(\dot{H}^{k}({\mathbb {T}})\) of the solution of the DNLS equation. We show that a functional measure on \(L^2({\mathbb {T}})\), absolutely continuous w.r.t. the Gaussian measure with covariance \(({\mathbb {I}}+(-\varDelta )^{k})^{-1}\), is associated to each integral of motion \({\textstyle \int }h_{2k}\), \(k\ge 1\).     

Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrödinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The L ^p and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrödinger type operators H ₂ are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderón-Zygmund operator if V?∈?B _2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x)?≤?Cm(x,V)².     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = |\nabla u|^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\). Here \(\Delta _p\), \(p>1\), is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)\), and the datum \(\sigma \) is a signed distribution in \(\Omega \). The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \(|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\), a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

for some \(C>0\). This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \). We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \(|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\). As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\), we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\), \(v\ge 0\) in \(\Omega \), and \(v=1\) on \(\partial \Omega \), which is interesting in its own right.

     

Order property and modulus of continuity of weak KAM solutions

For the Hamilton–Jacobi equation \(H(x,\partial _xu+c)=\alpha (c)\) with \(x\in \mathbb {T}^2\), it is shown in this paper that, for all \(c\in \alpha ^{-1}(E)\) with \(E>\min \alpha \), the elementary weak KAM solutions can be parameterized so that they are \(\frac{1}{3}\)-Hölder continuous in \(C^0\)-topology.     

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

The Riemann–Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding \(3\times 3\) matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Integrability and Linear Stability of Nonlinear Waves

It is well known that the linear stability of solutions of \(1+1\) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general \(N\times N\) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for \(N=3\) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.     

The Hardy–Schrödinger operator with interior singularity: the remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem \(L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}\) on a smooth bounded domain \(\Omega \) in \({\mathbb {R}}^n\) (\(n\ge 3\)) having the singularity 0 in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\), \(0\le s <2\), \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\le \lambda <\lambda _1(L_\gamma )\), the latter being the first eigenvalue of the Hardy–Schrödinger operator \(L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}\). There is a threshold \(\lambda ^*(\gamma , \Omega ) \ge 0\) beyond which the minimal energy is achieved, but below which, it is not. It is well known that \(\lambda ^*(\Omega )=0\) in higher dimensions, for example if \(0\le \gamma \le \frac{(n-2)^2}{4}-1\). Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when \( \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}\), to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of \(\Omega \) and \(\gamma \). If either \(s>0\) or if \(\gamma > 0\), i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega \), a notion that we introduce herein. On the other hand, and just like the case when \(\gamma =s=0\) studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), \(n=3\) is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when \(s=0\), \(\gamma \le 0\).     

On limit theorems for Banach-space-valued linear processes

Let \( {\left( {{\epsilon_i}} \right)_{i \in \mathbb{Z}}} \) be i.i.d. random elements in a separable Banach space \( \mathbb{E} \), and let \( \mathop {\left( {{a_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) be continuous linear operators from \( \mathbb{E} \) to a Banach space \( \mathbb{F} \) such that \( \sum\nolimits_{i \in \mathbb{Z}} {\left\| {{a_i}} \right\|} \) is finite. We prove that the linear process \( \mathop {\left( {{X_n}} \right)}\nolimits_{n \in \mathbb{Z}} \) defined by \( {X_n}: = \sum\nolimits_{i \in \mathbb{Z}} {{a_i}} \left( {{\epsilon_{n - i}}} \right) \) inherits from \( \mathop {\left( {{\epsilon_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) the central limit theorem and functional central limit theorems in various Banach spaces of \( \mathbb{F} \)-valued functions, including Hölder spaces.     

Differentiability of Spectral Functions for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Processes with Application to Large Deviations

We consider the relativistic α-stable process, a pure jump Markov process generated by \(\mathcal{H}^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha /2}-m\). Let ?C(λ) be the bottom of spectrum of Schrödinger type operator \(\mathcal{H}^{\lambda \mu} = \mathcal{H}^{\alpha} - \lambda \mu\), where μ is a signed Kato measure. We prove the differentiability of C(λ). As an application of it, we establish a large deviation principle for the additive functional \(A_t^{\mu}\) corresponding to the measure μ.     

On left-invariant Hörmander operators in $ {\mathbb{R}^N} $. applications to Kolmogorov–Fokker–Planck equations

If \( \mathcal{L} = \sum\limits_{j = 1}^m {X_j^2} + {X_0} \) is a Hörmander partial differential operator in \( {\mathbb{R}^N} \), we give sufficient conditions on the \( {X_{{j^{\text{S}}}}} \) for the existence of a Lie group structure \( \mathbb{G} = \left( {{\mathbb{R}^N},*} \right) \), not necessarily nilpotent, such that \( \mathcal{L} \) is left invariant on \( \mathbb{G} \). We also investigate the existence of a global fundamental solution Γ for \( \mathcal{L} \), providing results that ensure a suitable left-invariance property of Γ. Examples are given for operators \( \mathcal{L} \) to which our results apply: some are new; some have appeared in recent literature, usually quoted as Kolmogorov–Fokker–Planck-type operators. Nontrivial examples of homogeneous groups are also given.     

Products of Functions in $$H^1_{\rho }({{\mathcal {X}}})$$ and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$BMOρ(X) over RD-Spaces and Applications to Schrödinger Operators

Let \(({{\mathcal {X}}},d,\mu )\) be an RD-space, \(H^1_{\rho }({{\mathcal {X}}})\), and \({\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\) be, respectively, the local Hardy space and the local BMO space associated with an admissible function \(\rho \). Under an additional assumption that there exists a specific generalized approximation of the identity, the authors prove that the product \(f\times g\) of \(f\in H^1_{\rho }({{\mathcal {X}}})\) and \(g\in {\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\), viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from \(H^1_{\rho }({{\mathcal {X}}})\times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(L^1({{\mathcal {X}}})\) and from \(H^1_{\rho }({{\mathcal {X}}}) \times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(H^{\log }({{\mathcal {X}}})\), which is of wide generality. The authors also give out four applications of this result to Schrödinger operators, respectively, over different underlying spaces, where three of these applications are new.     

Generalized Frobenius partitions with 6 colors

We present the generating function for \(c\phi _6(n)\), the number of generalized Frobenius partitions of \(n\) with \(6\) colors, in terms of Ramanujan’s theta functions and exhibit \(2\), and \(3\)-dissections of it that yield the congruences \(c\phi _6(2n+1)\equiv 0~(\text {mod}~4)\), \(c\phi _6(3n+1)\equiv 0~(\text {mod}~3^2)\) and \(c\phi _6(3n+2)\equiv 0~(\text {mod}~3^2)\).     

Infinitely many congruences modulo 5 for 4-colored Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the family of functions \(c\phi _k(n),\) which denotes the number of generalized Frobenius partitions of \(n\) into \(k\) colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for \(c\phi _4(n)\) relative to different moduli. In this paper, employing classical results in \(q\)-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations, we prove a characterization of \(c\phi _4(10n+1)\) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by \(c\phi _4.\) This work greatly extends the recent work of Xia on \(c\phi _4\) modulo 5.