ANR project SMOOTH

2023-2027

Context of the project:

There has been in the last thirteen years considerable progresses in our understanding of some classes of random partial differential equations (PDEs) after the 2008 work of Burq & Tzvetkov on supercritical wave equations with random initial conditions and the 2014 works of Hairer and Gubinelli, Imkeller & Perkowski on singular stochastic elliptic and parabolic PDEs. These fundamental developments have laid the basis of a robust solution theory for equations that are classically ill-posed, due to a low regularity random data in the problem. While the regularity structures and paracontrolled approaches to stochastic PDEs have their roots in T. Lyons’ theory of rough paths, the progresses in the dispersive side have their roots in Bourgain’s works on invariant Gibbs measures for the nonlinear Schr ̈odinger equation in the mid nineties. In recent years, it became obvious that some of the conceptual ideas introduced in in the dispersive setting echo the strategy adopted to study singular stochastic PDEs. For instance, in both situations, any potential solution is decomposed as the sum of a ‘finite dimensional’ singular part enjoying regularity properties beyond the deterministic analysis, and a remainder part that is more regular and can be treated using purely analytical tools adapted to the equation – Bourgain spaces, regularity or paracontrolled structures. This project aims at a deeper clarification of the common features and key differences between the analysis of dispersive PDEs with random initial data and the analysis of singular stochastic PDEs. The objective will also consist in identifying new possible interactions between these do- mains, as well as developing specific tools for these classes of equations. The project will build upon ongoing collaborations between members of the dispersive and singular stochastic PDEs communities, and it will undoubtedly foster new ones.

Team members:

I. Bailleul (P.I.), N. Burq, V. Dang, A. Deya, C. Labbe, T. Robert, L. Thomann, N. Tzvetkov (co-P.I.), C. Sun L. Zambotti,

Activity:

Octobre 2023 Meeting in Nancy

May 2024 Meeting in Brest: Talks by Alberto, Alexis, Jiasheng, Aurelien, Van Duong, Yueh-Sheng, Cyril, Jacob, Leonard, Yvain and Laurent.

Publications:

  • Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space
    A. Debussche, R. Liu, N. Tzvetkov, N. Visciglia

  • New non-degenerate invariant measures for the Benjamin-Ono equation
    N. Tzvetkov

  • Blow-up for the 1D cubic NLS
    V. Banica, R. Lucà, N. Tzvetkov, L. Vega

  • Φ43 measures on compact Riemannian 3-manifolds
    I. Bailleul, N.V. Dang, L. Ferdinand, T.D. Tô

  • Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schrödinger equation
    N. Burq, A. Poiret, L. Thomann

  • Mean field singular stochastic PDEs
    I. Bailleul, N. Moench

  • Dynamics of quintic nonlinear Schrödinger equations in H2/5+(𝕋)
    J. Bernier, B. Grébert, T. Robert

  • Uniqueness of the Φ43 measures on closed Riemannian 3-manifolds
    I. Bailleul

  • Semi-classical observation suffices for observability: wave and Schrödinger equations
    N. Burq, B. Dehman, J. Le Rousseau

  • Post-Lie algebras of derivations and regularity structures
    J.-D. Jacques, L. Zambotti

  • Measure propagation along 𝒞0-vector field and wave controllability on a rough compact manifold
    N. Burq, B. Dehman, J. Le Rousseau

  • Wilson-Itô diffusions
    I. Bailleul, I. Cheryvev, M. Gubinelli

  • Quasi-invariance of Gaussian measures for the 3d energy critical nonlinear Schrödinger equation
    C. Sun, N. Tzvetkov

  • A remark on randomization of a general function of negative regularity
    T. Oh, M. Okamoto, O. Pocovnicu, N. Tzvetkov

  • Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of Hörmander's operators
    N. Burq, C. Zuily

  • Quantitative observability for one-dimensional Schrödinger equations with potentials
    P. Su, C. Sun, X. Yuan

  • On the 1d stochastic Schrödinger product
    A. Deya

  • Random models on regularity-integrability structures
    I. Bailleul, M. Hoshino

  • Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic Φ43-mode
    R. Liu, N. Tzvetkov, Y. Wang

  • Spectrally cut-off GFF, regularized Φ4 measure, and reflection positivity
    I. Bailleul, N.V. Dang, L. Ferdinand, G. Leclerc, J. Lin

  • Anderson stochastic quantization equation
    H. Eurly, A. Mouzard, T. Robert

  • Construction and spectrum of the Anderson Hamiltonian with white noise potential on R2 and R3
    Y.-S. Hsu, C. Labbé

  • Cutoff phenomenon in nonlinear recombinations
    P. Caputo, C. Labbé

  • On the stability of the Abrikosov lattice in the Lowest Landau Level
    P. Germain, V. Schwinte, L. Thomann

  • Convergence of dynamical stationary fluctuations
    C. Labbé, B. Laslier, F. Toninelli, L. Zambotti

  • Non-local quasilinear singular SPDEs
    I. Bailleul, H. Eulry

  • Probabilistic well-posedeness for the nonlinear Schrödinger equation on the 2d sphere I: positive regularities
    N. Burq, N. Camps, Ch. Sun, N. Tzvetkov

  • The Second Picard iteration of NLS on the 2d sphere does not regularize Gaussian random initial data
    N. Burq, N. Camps, M. Latocca, Ch. Sun, N. Tzvetkov

  • Almost surely smoothed scattering for cubic NLS
    N. Burq, H. Koch, N. Visciglia, N. Tzvetkov

  • Measure and continuous vector field at a boundary I: propagation equations and wave observability
    N. Burq, B. Dehman, J. Le Rousseau

  • Measure and continuous vector field at a boundary II: geodesics and support propagation
    N. Burq, B. Dehman, J. Le Rousseau

  • The Cauchy problem for the periodic Kadomtsev-Petviashvili-II equation below L2
    S. Herr, R. Schippa, N. Tzvetkov

  • Delocalized eigenvectors of transitive graphs and beyond
    N. Burq, C. Letrouit

  • New bounds on the high Sobolev norms of the 1d NLS solutions
    D. Berti, F. Planchon, N. Tzvetkov, N. Visciglia

  • Renormalization of random models: a review
    I. Bailleul, M. Hoshino

  • Remarks on nonlinear dispersive PDEs with rough dispersion management
    T. Robert

  • Regularization by noise for some strongly non-resonant modulated dispersive PDEs
    T. Robert

  • Nonlinear interpolation and the flow map for quasilinear equations
    Th. Alazard, N. Burq, M. Ifrim, D. Tataru, C. Zuily

  • Strichartz Estimates for a Class of Baouendi-Grushin Operators
    N. Burq, M. Latocca

  • Local well-posedness for the periodic Boltzmann equation with constant collision kernel
    E. Basakoglu, N. Tzvetkov, Ch. Sun, Y. Wang

  • Dirac operators and local invariants on perturbations of Minkowski space
    N.V. Dang, A. Vasy, M. Wrochna

  • Infinite dimensional invariant tori for nonlinear Schrödinger equations
    J. Bernier, B. Grebert, T. Robert

  • Local expansion properties of paracontrolled systems
    I. Bailleul, N. Moench

  • Stochastic complex Ginzburg-Landau equation on compact surfaces
    T. Robert, Y. Zine

  • On the parabolic Phi43 model for the harmonic oscillator: diagrams and local existence
    A. Deya, R. Fukuizumi, L. Thomann

  • Global results for weakly dispersive KP-II equations on the cylinder
    S. Herr, R. Schippa, N. Tzvetkov

  • Hyperbolic nonlinear Schrödinger equations on R×T
    E. Başakoğlu, C. Sun, N. Tzvetkov, Y. Wang

  • Asymptotics of zeta determinants of Laplacians on large degree abelian covers
    N.V. Dang, J. Lin, F. Naud

  • Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials
    G. Cannizzaro, C. Labbe, W. van Zuijlen

  • On low regularity well-posedness of the binormal flow
    V. Banica, R. Luca, N. Tzvetkov, L. Vega

  • Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
    N. Burq, P. Germain, M. Sorella, H. Zhu

  • Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
    G. Li, J. Li, T. Oh, N. Tzvetkov

  • Dynamical interface above a hard wall and reflected SPDE on the half-line
    P. Faugere, C. Labbe

  • Poincare series for analytic convex bodies
    N.V. Dang, Y.Guedes Bonthonneau, M. Leautaud, G. Riviere

  • Observability of Schrödinger propagators on tori in rough settings
    N. Burq, H. Zhu

  • Resolvent bounds imply observability from measurable time sets for Schrödinger equations
    N. Burq, H. Zhu

  • Local wellposedness of the 2d Anderson-Gross-Pitaevskii equation
    S. Mackowiak

  • Large torus limit of global dynamics of the two-dimensional dispersive Anderson model
    R. Liu, N. Tzvetkov

  • Transportation cost inequalities for singular SPDEs
    I. Bailleul, M. Hoshino, R. Takano

  • Almost sure global nonlinear smoothing for the 2D NLS
    C. Sun, N. Tzvetkov

  • Five lectures on regularity structures and SPDEs
    I. Bailleul

  • A measure L infinity div-curl lemma
    V. Banica, N. Burq

  • Standing waves of the Anderson-Gross-Pitaevskii equation
    S. Mackowiak

  • The Yang--Mills measure on compact surfaces as a universal scaling limit of lattice gauge models
    N. V. Dang, E. Nohra

  • The Yang--Mills measure on compact surfaces as a universal scaling limit of lattice gauge models
    N. V. Dang, E. Nohra

  • An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding
    Y.C. Huang, T. Ozawa, Ch. Sun, T. Takeuchi

  • Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the 3d ball
    N. Burq, N. Camps, Ch. Sun, N. Tzvetkov

  • Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations
    W. Ruffenach, N. Tzvetkov

  • On the pointwise convergence of NLS flow on S2
    F. Meng, Y. Song, Ch. Sun, R. Zhang, J. Zheng

  • On probabilistic ill-posedness
    T. Oh, N. Tzvetkov

  • Remainders of generalised Taylor expansions and a priori bounds for rough differential equations
    L. Agabiti, A. Bonicelli, L. Zambotti

  • The Yang-Mills measure on surfaces via Morse theory
    R. Chhaibi, N. V. Dang, Y. Guedes Bonthonneau, G. Riviere, T.D. To

  • A simple derivation of Anderson-Hartree equations with harmonic potential
    S. Mackowiak

  • On the parabolic Φ43 model for the harmonic oscillator II: global existence and invariant measures
    A. Deya, R. Fukuizumi, L. Thomann

  • From skew to reflected SPDEs
    C. Labbe, Th. Le Guerch, L. Zambotti

  • On the stability of solutions to the 3-D anisotropic Navier-Stokes equations in the critical space
    N. Burq, N. Liu, P. Zhang

  • Dirac measures can be quantum limits on the Sierpinski gasket
    P. Alonso Ruiz, N. Tzvetkov

  • Stochastic PDE construction of some non-local or non-translation-invariant Anderson operators in the full subcritical regime
    L Ferdinand, S. Mackowiak, A. Minguella

  • Long time dynamics for complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds
    X. Cheng, Z. Lyu, Ch. Sun, Y. Zheng