Small Cauchy Data Research Articles

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain’s Lemma which provides a partition of the “resonant sites” of the Laplace operator on irrational tori.

Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaces

We study the Cauchy problems for the Klein-Gordon (HNLKG), wave (HNLW), and Schrodinger (HNLS) equations with cubic convolution (of Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrodinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. As a consequence, we get local and global well-posedness and scattering in larger than usual \(L^p\)-Sobolev spaces. For more information see https://ejde.math.txstate.edu/Volumes/2021/101/abstr.html

Koopman operator for Burgers's equation

We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).

Long time existence for fully nonlinear NLS with small Cauchy data on the circle

In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schr\"odinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\varepsilon$ sufficiently small, the lifespan of the solution is of order $\varepsilon^{-N}$ for any $N\in\mathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.

Global existence of Landau–Lifshitz–Gilbert equation and self-similar blowup of Harmonic map heat flow on [formula omitted

Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau–Lifshitz–Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on S2. As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss

In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker–Pierfelice and Ionescu–Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated/microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the L2 well-posedness and L2 scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data.

On global classical solutions of the three dimensional relativistic Vlasov-Darwin system

We study the Cauchy problem of the relativistic Vlasov-Darwin system with generalized variables proposed by Sospedra-Alfonso et al. [“Global classical solutions of the relativistic Vlasov-Darwin system with small Cauchy data: the generalized variables approach,” Arch. Ration. Mech. Anal. 205, 827-869 (2012)]. We prove global existence of a non-negative classical solution to the Cauchy problem in three space variables under small perturbation of the initial datum, and as a consequence, we obtain that nearly spherically symmetric solutions with required regularity exist globally in time.

Analytic smoothing effect for nonlinear Schrödinger equation with quintic nonlinearity

We prove the global existence of analytic solutions to nonlinear Schrödinger equation with quintic nonlinearity in n space dimensions for sufficiently small Cauchy data with exponential decay. The smallness assumption on the data is imposed in terms of the critical Sobolev space H˙n/2−1/2. Moreover, a characterization of some class of analytic functions is given.

Global Classical Solutions of the Relativistic Vlasov–Darwin System with Small Cauchy Data: The Generalized Variables Approach

We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system. Our proof extends the method used by Loeper in J. Math. Pures Appl. 86, 68-79 (2006) to obtain uniqueness results for the Vlasov-Poisson system.

Condition de non-résonance pour l’oscillateur harmonique quantique perturbé

On R d , we consider the differential operator −�+jj xjj +R(x)+M. Here R is in the Schwartz class and M is a bounded operator of L (R). Under a localization hypothesis on the spectrum of −� + jj xjj + R(x), which is automatically satisfied if d = 1, one proves that there are arbitrarily small operator M, i.e. jj M jj � 1, such that for all r � 3, for small Cauchy data of order in high Sobolev norms, the nonlinear Schrodinger equation admits a solution which is bounded by 2 on a time existence interval of length � −r . The case R = 0 has been proved by Grebert-Imekraz-Paturel. Here we do not need any explicit spectral analysis (eigenvalues and eigenfunctions) of −� + jj xjj + R(x). The role of M is making the spectrum of −�+ jj xjj + R(x)+ M nonresonant. We also give a partial answer in the case M = 0 : there are some potentials R such that the spectrum of −@2 x + x2 + R(x) is nonresonant. For this, we use some Chelkak-Kargaev-Korotyaev's results about the inverse spectral problem of harmonic oscillator and construct explicit dual basis of the squared Hermite functions. Sur R d , nous considerons l'operateur differentiel −� + jj xjj + R(x) +

Normal forms for semilinear superquadratic quantum oscillators

On the real line, we consider nonlinear Hamiltonian Schrödinger equations with the superquadratic oscillator − d 2 / d x 2 + x 2 p + η ( x ) + M , where p is an integer ⩾2, η is a polynomial of degree < 2 p such that inf ( x 2 p + η ( x ) ) ⩾ 0 , and M is a multiplier (i.e. simultaneously diagonalized with − d 2 / d x 2 + x 2 p + η ( x ) ). A previous article (Grébert et al. (2009) [11]) contains the case p = 1 in R d . Here we deal with d = 1 but we authorize any superquadratic potential. Under generic conditions on M related to the nonresonance of the linear part, such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order. Consequently we deduce long time existence for solutions of the above equation with small Cauchy data in the high Sobolev spaces. As spectral analysis (spectrum and eigenfunctions) of the linear part is not explicit, we use Helffer–Robert and Yajima–Zhangʼs results (Helffer and Robert (1982) [12], Yajima and Zhang (2001) [21]) to understand asymptotic behavior of both spectrum and eigenfunctions.

Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +{|x|}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \mathbb{R}^{d},\, t\in \mathbb{R},$$where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.

Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle

We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on S 1 \mathbb S^1 remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.

Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds

AbstractThis paper is devoted to the proof of almost global existence results for Klein‐Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the Laplacian perturbed by a potential on Zoll manifolds. © 2007 Wiley Periodicals, Inc.

The global Cauchy problem for the NLS and NLKG with small rough data

By using the unit-cube decomposition to the frequency spaces, we study the Cauchy problem for the nonlinear Schrödinger equation and the nonlinear Klein–Gordon equation. Some global well posedness results are obtained for the small Cauchy data in some modulation spaces M p , 1 s .

Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds

We prove a long time existence result for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds. This generalizes a preceding result concerning the case of spheres, obtained in an earlier paper by the authors. The proof relies on almost orthogonality properties of products of eigenfunctions of positive elliptic selfadjoint operators on a compact manifold and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

Remarks on modified improved Boussinesq equations in one space dimension

We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term behaving as a power as . Solutions are considered in space for all . According to the value of s , the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J . 45 , 797–816) obtained the minimum value of p greater than 8 at for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than at and the corresponding solution u has the time decay, such as as . We also prove non-existence of non-trivial asymptotically free solutions for under vanishing condition near zero frequency on asymptotic states.

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation \(u_{tt} - \Delta u = f(u)\) with \(f(u) = \gamma |u|^{\alpha+1}\) or \(f(u) = \gamma|u|^{\alpha}u\) in R 3×R + for small Cauchy data is proven if \(\sqrt{2} < \alpha < 2\). A counterexample is given which shows that the lower bound on α is sharp.

Nonlinear Schrödinger Equations in the Sobolev Space of Critical Order

The Cauchy problem for the nonlinear Schrödinger equations is considered in the Sobolev spaceHn/2(Rn) of critical ordern/2, where the embedding intoL∞(Rn) breaks down and any power behavior of interaction works as a subcritical nonlinearity. Under the interaction of exponential type the existence and uniqueness is proved for globalHn/2-solutions with small Cauchy data.