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On a fractional nonlinear Schrödinger equation with irregular coefficients. case: d<2s

Arshyn Altyby, Michael Ruzhansky, Mohammed Elamine Sebih, Niyaz Tokmagambetov

Abstract

In the case when $d<2s$, where $d$ is the space dimension and $s$ is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schrödinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: $d=1, s=1$. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.

On a fractional nonlinear Schrödinger equation with irregular coefficients. case: d<2s

Abstract

In the case when , where is the space dimension and is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schrödinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: . A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.
Paper Structure (16 sections, 7 theorems, 97 equations, 4 figures)

This paper contains 16 sections, 7 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Duhamel's principle
  4. Gronwall's type inequalities
  5. Energy estimates for the classical solution
  6. Very weak well-posedness
  7. Existence of very weak solutions
  8. Uniqueness
  9. Compatibility with classical theory
  10. Numerical simulations
  11. Case 1 (regular reference).
  12. Case 2 (singular potential).
  13. Case 3 (singular source).
  14. Case 4 (both coefficients singular).
  15. Discussion.
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\Omega$ be a standard domain in $\mathbb{R}^d$. Then, for $r,s_1,s_2,p_1,p_2,q,\theta,d$ satisfying the inequality holds for all $f\in W^{s_1,p_1}(\Omega) \cap W^{s_2,p_2}(\Omega)$, with the following exceptions, when it fails:

Figures (4)

  • Figure 1: Case 1: real and imaginary parts of $u(t,x)$ at $t=10$.
  • Figure 2: Case 2: real and imaginary parts of $u(t,x)$ for $\varepsilon\in\{1.0,0.7,0.3,0.01\}$.
  • Figure 3: Case 3: real and imaginary parts of $u(t,x)$ for $\varepsilon\in\{0.015,0.01,0.009,0.005\}$.
  • Figure 4: Case 4: real and imaginary parts of $u(t,x)$ for $\varepsilon\in\{1.0,0.7,0.3,0.01\}$.

Theorems & Definitions (20)

  • Proposition 2.1: Fractional GNS inequality, e.g. Theorem 1. BM19
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 1: Moderateness
  • Remark 3.1
  • Definition 2: Very weak solution
  • Remark 3.2
  • ...and 10 more