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Global wellposedness of general nonlinear evolution equations for distributions on the Fourier half space

Kenji Nakanishi, Baoxiang Wang

TL;DR

This work develops a unified Fourier-space framework for global wellposedness of very general nonlinear evolution equations written with Fourier multipliers and analytic nonlinearities. By introducing the distribution-space, half-space-support space $\mathcal{X}(\mathrm{v})$ (and its cone-restricted variants $\mathcal{X}_{\mathcal{R}}(\mathrm{v})$), the authors prove global existence without size restrictions on rough initial data, while allowing arbitrarily strong growth compatible with Fourier support. The approach relies on time-dependent Fourier weights and careful convolution bounds to control nonlinear frequency transfer, yielding a contraction mapping in a refined function space and extending wellposedness to broad classes of PDEs, including Navier–Stokes, Euler, nonlinear wave/Klein–Gordon, and nonlinear Schrödinger equations. The results also demonstrate optimality of the half-space and boundary-decay conditions, and provide simpler locally integrable-frequency variants and extensive examples, highlighting both theoretical depth and potential practical impact for rough data regimes in fluid dynamics and dispersive/hyperbolic systems.

Abstract

The Cauchy problem is studied for very general systems of evolution equations, where the time derivative of solution is written by Fourier multipliers in space and analytic nonlinearity, with no other structural requirement. We construct a function space for the Fourier transform embedded in the space of distributions, and establish the global wellposedness with no size restriction. The major restriction on the initial data is that the Fourier transform is supported on the half space, decaying at the boundary in the sense of measure. We also require uniform integrability for the orthogonal directions in the distribution sense, but no other condition. In particular, the initial data may be much more rough than the tempered distributions, and may grow polynomially at the spatial infinity. A simpler argument is also presented for the solutions locally integrable in the frequency. When the Fourier support is slightly more restricted to a conical region, the generality of equations is extremely wide, including those that are even locally illposed in the standard function spaces, such as the backward heat equations, as well as those with infinite derivatives and beyond the natural boundary of the analytic nonlinearity. As more classical examples, our results may be applied to the incompressible and compressible Navier-Stokes and Euler equations, the nonlinear diffusion and wave equations, and so on. The major drawback of the Fourier support restriction is that the solutions cannot be real valued.

Global wellposedness of general nonlinear evolution equations for distributions on the Fourier half space

TL;DR

This work develops a unified Fourier-space framework for global wellposedness of very general nonlinear evolution equations written with Fourier multipliers and analytic nonlinearities. By introducing the distribution-space, half-space-support space (and its cone-restricted variants ), the authors prove global existence without size restrictions on rough initial data, while allowing arbitrarily strong growth compatible with Fourier support. The approach relies on time-dependent Fourier weights and careful convolution bounds to control nonlinear frequency transfer, yielding a contraction mapping in a refined function space and extending wellposedness to broad classes of PDEs, including Navier–Stokes, Euler, nonlinear wave/Klein–Gordon, and nonlinear Schrödinger equations. The results also demonstrate optimality of the half-space and boundary-decay conditions, and provide simpler locally integrable-frequency variants and extensive examples, highlighting both theoretical depth and potential practical impact for rough data regimes in fluid dynamics and dispersive/hyperbolic systems.

Abstract

The Cauchy problem is studied for very general systems of evolution equations, where the time derivative of solution is written by Fourier multipliers in space and analytic nonlinearity, with no other structural requirement. We construct a function space for the Fourier transform embedded in the space of distributions, and establish the global wellposedness with no size restriction. The major restriction on the initial data is that the Fourier transform is supported on the half space, decaying at the boundary in the sense of measure. We also require uniform integrability for the orthogonal directions in the distribution sense, but no other condition. In particular, the initial data may be much more rough than the tempered distributions, and may grow polynomially at the spatial infinity. A simpler argument is also presented for the solutions locally integrable in the frequency. When the Fourier support is slightly more restricted to a conical region, the generality of equations is extremely wide, including those that are even locally illposed in the standard function spaces, such as the backward heat equations, as well as those with infinite derivatives and beyond the natural boundary of the analytic nonlinearity. As more classical examples, our results may be applied to the incompressible and compressible Navier-Stokes and Euler equations, the nonlinear diffusion and wave equations, and so on. The major drawback of the Fourier support restriction is that the solutions cannot be real valued.
Paper Structure (29 sections, 27 theorems, 281 equations)

This paper contains 29 sections, 27 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Global wellposedness results
  3. Solutions locally integrable in frequency
  4. Outline of the paper
  5. Notations
  6. Spaces of distributions for the Fourier transform of solutions
  7. Characterization of $\mathcal{X}$
  8. Local basis in $\mathcal{X}$
  9. Bounded sets in $\mathcal{X}$
  10. Compact sets in $\mathcal{X}$
  11. Convolution for the distributions
  12. Global wellposedness for distributions
  13. Comparison with other solutions
  14. Solutions beyond blow-up
  15. Solutions beyond the natural boundary
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $d\in\mathbb{N}$, $\mathrm{v}\in\mathbb{R}^d\setminus\{0\}$ and assume C-s. Then the Cauchy problem of GEE is globally wellposed in $(\mathcal{F}^{-1}\mathcal{X}(\mathrm{v}))^{n_1}$ on $t\ge 0$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 44 more