**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 Graph Search.

Publication# Probabilistic and Deterministic Wellposedness for Low Regularity Dispersive Equations

Abstract

In this thesis we will present two results on global existence for nonlinear dispersive equations with data at or below the scaling regularity. In chapter 1 we take a probabilistic perspective to study the energy-critical nonlinear Schrödinger equation in dimensions d>6. We prove that the Cauchy problem is almost surely globally well-posed with scattering for a range of super-critical initial data. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results in dimension 4 and the main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity. Chapter 2 concerns the half-wave maps equation, a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. We will prove that in three dimensions the equation is "weakly" globally well-posed for angularly regular data which is small in a critical Besov space, partially generalising known results in dimensions d=4. The main difficulty in moving to three dimensions is the loss of a key endpoint Strichartz estimate. We overcome this by using Sterbenz's improved Strichartz estimates in conjunction with commuting vector fields to develop trilinear estimates in weighted Strichartz spaces which avoid the use of the Strichartz endpoint.

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 publications (37)

Related MOOCs (32)

Related concepts (40)

Digital Signal Processing [retired]

The course provides a comprehensive overview of digital signal processing theory, covering discrete time, Fourier analysis, filter design, sampling, interpolation and quantization; it also includes a

Digital Signal Processing

Digital Signal Processing is the branch of engineering that, in the space of just a few decades, has enabled unprecedented levels of interpersonal communication and of on-demand entertainment. By rewo

Digital Signal Processing I

Basic signal processing concepts, Fourier analysis and filters. This module can
be used as a starting point or a basic refresher in elementary DSP

Big data

Big data primarily refers to data sets that are too large or complex to be dealt with by traditional data-processing application software. Data with many entries (rows) offer greater statistical power, while data with higher complexity (more attributes or columns) may lead to a higher false discovery rate. Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe big data is the one associated with a large body of information that we could not comprehend when used only in smaller amounts.

Data

In common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.

Global governance

Global governance refers to institutions that coordinate the behavior of transnational actors, facilitate cooperation, resolve disputes, and alleviate collective action problems. Global governance broadly entails making, monitoring, and enforcing rules. Within global governance, a variety of types of actors – not just states – exercise power. Governance is thus broader than government. Global governance began in the mid-19th century. It became particularly prominent in the aftermath of World War I, and more so after the end of World War II.

Katie Sabrina Catherine Rosie Marsden

We study the energy-critical nonlinear Schrodinger equation with randomised initial data in dimensions d > 6. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised supercritical initial data in H-s(Rd) wheneve ...

Christophe Marcel Georges Galland, Valeria Vento, Sachin Suresh Verlekar, Philippe Andreas Rölli

Data and code associated with the following paper: V. Vento et al, Nano Lett. 2023 A thorough explanation of the experiment performed is available there. The name of each sub-folder and file in Maps_data_code.zip indicates the correspondin ...

, , , , ,

Dataset corresponding to the following manuscript: Slagter, A., Setyadji, J.A., Vogt, E.L. et al. Nanoindentation Hardness and Modulus of Al2O3–SiO2–CaO and MnO–SiO2–FeO Inclusions in Iron. Metall Mater Trans A (2024). https://doi.org/10.1007/s11661-024-0 ...