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The zero-dispersion limit for the Benjamin--Ono equation on the circle

Ola Mæhlen

Abstract

Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle $\mathbb{T}= \mathbb{R}/2π\mathbb{Z}$ with bounded initial data $u_0\in L^\infty(\mathbb{T},\mathbb{R})$. The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with $u_0\in L^2\cap L^\infty(\mathbb{R})$. Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the branches of the multivalued solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate.

The zero-dispersion limit for the Benjamin--Ono equation on the circle

Abstract

Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle with bounded initial data . The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with . Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the branches of the multivalued solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate.

Paper Structure

This paper contains 12 sections, 12 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some history and related works
  3. Strategy and main results
  4. Preliminaries
  5. The Hardy spaces and Toeplitz operators
  6. The explicit formula and its zero-dispersion limit
  7. The generalization of Miller--Xu's alternating sum formula
  8. Proving the main results
  9. Characterizing the zero-dispersion limit
  10. Proving the Raney-type lemma
  11. Properties of the zero-dispersion limit
  12. Auxiliary lemmas

Key Result

Theorem 1.1

Let $u_0\in L^\infty(\mathbb{T},\mathbb{R})$ and $u^\varepsilon\in C(\mathbb{R}, L^2(\mathbb{T},\mathbb{R}))$ denote the unique solutionIn the sense of Molinet MolinetPeriodic or, equivalently, Gérard--Kappeler SharpWellPosednessCircle. of eq: BOepsilon. Then, as $\varepsilon\to 0$, $u^\varepsilon$ locally uniformly in $t\in\mathbb{R}$. The function $ZD[u_0](t)$ is, for a.e. $x\in \mathbb{T}$, gi

Figures (2)

  • Figure 1: The graph (black curve) of the multivalued solution $u_B=u_0(x-2t u_B)$ of Burgers' equation, shown at $t=0$ and at a later time $t>0$. The red region corresponds to $\chi(u_0(x-2ty),y)=1$, while the blue region corresponds to $\chi(u_0(x-2ty),y)=-1$. The dashed vertical line intersects the graph once initially and seven times at the later time; the alternating sum of these seven $y$-values coincides with the integral of $\chi(u_0(x-2ty),y)$ along the dashed line.
  • Figure 2: A visual representation of the twelve weights $S(T^j m,T^\ell n)$ resulting from $m=(0,1,1,1)$ and $n=(0,1,2)$. The diagram at column $j$ and row $\ell$ corresponds to $S(T^j m,T^\ell n)$. The red and blue line represents the partial sums of $T^j m$ and $T^\ell n$ respectively; the former grows vertically and the latter horizontally. Moreover, $S(T^j m,T^\ell n)=1$ if the red line is below the blue, and $S(T^j m,T^\ell n)=0$ otherwise, as justified by Lemma \ref{['lem: aSimplerCharacterizationOfWhenTheCompositionOfShiftsIsOne']}. Here, we see that $S(T^j m,T^\ell n)=1$ when $(j,\ell)\in\{2,3,4\}\times\{3\}$.

Theorems & Definitions (30)

  • Theorem 1.1: Characterizing the zero-dispersion limit
  • Theorem 1.2: Properties of $ZD{[\cdot]}$
  • Lemma 2.1: Compositions of Toeplitz-shifts
  • proof
  • Proposition 2.2: Existence of a zero-dispersion limit
  • proof
  • Remark 2.3: On the restriction $u_0\in L^\infty(\mathbb{T},\mathbb{R})$
  • Remark 2.4: Letting the initial data depend on $\varepsilon$
  • Proposition 2.5: Generalized alternating sum formula
  • Remark 2.6
  • ...and 20 more