First accessible work giving an exhaustive and up-to-date presentation of how to use Fourier analysis to study PDEs. Written by experts in the field of Fourier analysis, this work presents a self-contained state of the art of techniques with applications to different classes of PDEs. Both acce...
In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDEs. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics eq...
A Fourier Analysis Approach to Elliptic Equations with Critical Potentials and Nonlinear Derivative TermsNonlinear elliptic equationssingular potentialsfractional derivativessymmetryWe study nonhomogeneous elliptic problems considering a general linear elliptic operator with singular critical potentials and ...
The layer processes the input by applying convolutions and spectral convolutions and sums the outputs. This diagram illustrates the architecture of the layer. A spectral convolutional layer applies convolutions in the frequency domain and is particularly useful for solving PDEs as it allows the networ...
(PDE). Evidently, there is no universal theory for the solution of nonlinear PDEs, but there exists a distinguished class of nonlinear equations that can be solved with a mathematical rigour: the so-calledintegrable systems. The history of integrable PDEs started in the 1960s when Gardner et ...
In this chapter, we introduce the theory of Fourier transforms and use it to find solutions of PDEs on infinite domains. For example, we consider in some detail the problem of heat conduction in an infinite rod: * $$D.E.{u_t} = k{u_{xx}} - \infty 0,k > o,I.C.u(x,0) =...
It is well known that a large class of PDEs can be written as a Hamiltonian PDE (1)Mzt+Kzx=∇zS(z), where z(x,t)∈Rn(n≥3),M and K are skew-symmetric matrices, and S(z) is a smooth Hamiltonian function. These PDEs include the KdV equation [1], nonlinear Schrödinger equati...
The rise of deep learning techniques has opened new opportunities for accelerating the modeling of complex nonlinear interactions and system dynamics. Recently, the Fourier Neural Operator (FNO) has been shown to be very promising in accelerating solving the Partial Differential Equations (PDEs) and ...
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To simplify our analysis, we will assume g = 0 in (1.2), although this can be treated in a similar manner as shown in [11]. 2.1.1. Phase space and Young measure. We first introduce some notation: • C0(Rd) denotes the closure under the supremum norm of compactly sup- ported, ...