**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# On long time behavior of solutions to nonlinear dispersive equations

Abstract

In part I, we address the issue of existence of solutions for Cauchy problems involving nonlinear hyperbolic equations for initial data in Sobolev spaces with scaling subcritical regularity. In particular, we analyse nonlinear estimates for null-forms in the context of wave Sobolev spaces $H^{s,b}$, first in a flat background, then we generalize to more general curved backgrounds. We provide the foundations to show that the Yang-Mills equation in $\mathbb{R}^{1+3}$ are globally well-posedness for small weighted $H^{3/4+}\times H^{-1/4+}$ initial data, matching the minimal regularity obtained by Tao \cite{tao03}. Our method, inspired from \cite{dasgupta15}, combines the classical Penrose compactification of Minkowski space-time with a null-form estimates for second order hyperbolic operators with variable coefficients. The proof of the null-form appearing in the Yang-Mills equation will be provided in a subsequent work. As a consequence of our argument, we shall obtain sharp pointwise decay bounds.

In part II, we show that the finite time type II blow-up solutions for the energy critical nonlinear wave equation $\Box u = -u^5$ on $\R^{3+1}$ constructed in \cite{krieger2009slow}, \cite{krieger2009renormalization} are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $\lambda(t) = t^{-1-\nu}$ is sufficiently close to the self-similar rate, i. e. $\nu>0$ is sufficiently small. This result is qualitatively optimal in light of the result of \cite{krieger2015center}, it builds on the analysis of \cite{krieger2017stability} and it is joint work with my thesis advisor Prof. J. Krieger.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related concepts (15)

Related publications (3)

Perturbation theory

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A c

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function

Loading

Loading

Loading

Stefano Francesco Burzio, Joachim Krieger

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation [ \Box u = -u^5 ] on $\R^{3+1}$ constructed in \cite{KST}, \cite{KS1} are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $\lambda(t) = t^{-1-\nu}$ is sufficiently close to the self-similar rate, i. e. $\nu>0$ is sufficiently small. This result is qualitatively optimal in light of the result of \cite{CNLW4}. The paper builds on the analysis of \cite{CondBlow}.

2018Francesca Bonizzoni, Fabio Nobile

We study the Darcy boundary value problem with lognormal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Hölder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived.

2020Francesca Bonizzoni, Fabio Nobile

We study the Darcy boundary value problem with log-normal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Hölder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived.