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A new approach for the analysis of evolution partial differential equations on a finite interval

Türker Özsarı, Dionyssios Mantzavinos, Konstantinos Kalimeris

TL;DR

This work develops a general framework to analyze evolution PDEs on a finite interval by reducing the problem to associated half-line problems via the unified transform (Fokas method). It derives explicit integral equations for the half-line boundary data and proves local well-posedness in Sobolev spaces for canonical linear models, notably the heat equation and the linear KdV equation, including extensions to time-dependent coefficients. The key contributions are the finite-interval-to-half-line reductions, the explicit integral equations for the boundary data, and the rigorous Sobolev-space solvability results (L^2, H^1, H^m) achieved through contraction mappings and interpolation techniques, enabling a pathway to nonlinear well-posedness from linear estimates. The approach provides a versatile tool to transfer regularity and well-posedness from half-line problems to finite-interval settings, with potential broad applicability to nonlinear problems and to a wider class of evolution equations.

Abstract

We show that, for certain evolution partial differential equations, the solution on a finite interval $(0,\ell)$ can be reconstructed as a superposition of restrictions to $(0,\ell)$ of solutions to two associated partial differential equations posed on the half-lines $(0,\infty)$ and $(-\infty,\ell)$. Determining the appropriate data for these half-line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed point argument in $L^2$-based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg-de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time-dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half-line. These results can, in turn, be used to establish local well-posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks.

A new approach for the analysis of evolution partial differential equations on a finite interval

TL;DR

This work develops a general framework to analyze evolution PDEs on a finite interval by reducing the problem to associated half-line problems via the unified transform (Fokas method). It derives explicit integral equations for the half-line boundary data and proves local well-posedness in Sobolev spaces for canonical linear models, notably the heat equation and the linear KdV equation, including extensions to time-dependent coefficients. The key contributions are the finite-interval-to-half-line reductions, the explicit integral equations for the boundary data, and the rigorous Sobolev-space solvability results (L^2, H^1, H^m) achieved through contraction mappings and interpolation techniques, enabling a pathway to nonlinear well-posedness from linear estimates. The approach provides a versatile tool to transfer regularity and well-posedness from half-line problems to finite-interval settings, with potential broad applicability to nonlinear problems and to a wider class of evolution equations.

Abstract

We show that, for certain evolution partial differential equations, the solution on a finite interval can be reconstructed as a superposition of restrictions to of solutions to two associated partial differential equations posed on the half-lines and . Determining the appropriate data for these half-line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed point argument in -based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg-de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time-dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half-line. These results can, in turn, be used to establish local well-posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks.

Paper Structure

This paper contains 14 sections, 6 theorems, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Heat Equation
  3. An integral equation for $a(t)$
  4. Existence in $L^2(0, T)$
  5. Existence in $H^1(0, T)$
  6. Existence in $H^m(0, T)$ for any $m\geq 0$
  7. The Linear K dV Equation
  8. An integral equation for $a(t)$
  9. Existence in $L^2(0, T)$
  10. Existence in $H^m(0, T)$ for any $m\geq 0$
  11. Concluding remarks
  12. Uniqueness and global solvability
  13. Numerical illustrations
  14. Evolution equations with time-dependent coefficients

Key Result

Theorem 1

Let $m \geq 0$, $\ell>0$ and $0 < T \leq \dfrac{\sqrt \pi e^{\frac{3}{2}}}{2 \cdot 3^{\frac{5}{4}}} \, \ell^2$. Then, for Dirichlet boundary data $g \in H^m(0, T)$ such that $g^{(n)}(0) = 0$ for all integers $0\leq n < m-\frac{1}{2}$, there exist functions $a, b \in H^m(0, T)$ such that the solution can be expressed as the sum of the restrictions on $(0, \ell)$ of the solutions to the heat equati

Figures (4)

  • Figure 2.1: The region $\mathcal{D}$ and its positively oriented boundary $\partial \mathcal{D}$.
  • Figure 3.1: The oriented contours $\mathcal{C}_+$ (left) and $\mathcal{C}_-$ (right) for the solution formulae \ref{['v-kdv-sol']} and \ref{['w-kdv-sol']}, as defined by \ref{['cpm-kdv']}.
  • Figure 4.1: Top panel: Evaluation of the unified transform formulae \ref{['heat-utm-t']} and \ref{['w-utm-t']} for the solutions $v(x,t)$ (red) and $w(x,t)$ (blue) of the half-line problems \ref{['vw-hl-ibvp-i']} in the case of the boundary data \ref{['ab-num']}, which are obtained via the numerical solution of the integral equation \ref{['a-int-eq']} for $g(t) = \sin\left(2\pi t/\ell^2\right)$ supported for $t\in(0,T)$ with $T=3 \ell^2/8$. The surface colored in green corresponds to the sum $v(x,t)+w(x,t)$ and is virtually indistinguishable from the surface obtained by plotting the unified transform formula \ref{['q-sol']} for the solution $q(x,t)$ to the finite interval problem \ref{['q-fi-ibvp-i']} with boundary datum $g(t)$, thus verifying the decomposition \ref{['fi-hl-dec-i']}. Bottom panels: The discrepancy $dc(x,t)=q(x,t)-\left[v(x,t)+w(x,t)\right]$ for $(x,t)\in (0,\ell)\times(0,T)$ with $\ell=4$ and $T = 6 < 4^2=\ell^2$, in line with the contraction condition \ref{['T-L2']} (bottom left), and the norm $\left\| dc(t) \right\|_{L_x^2(0,\ell)}$ for $t\in (0,T)$ in logarithmic scale (bottom right). Perfect agreement is observed between the numerical evaluation of the formulae for $q$ and $v+w$, consistent with the fact that the corresponding surfaces coincide in the top panel (green).
  • Figure 4.2: Same setup with the one of Figure \ref{['fig:3D-ab-s']} but for $g(t) = \sin\left(2\pi t/(7\ell^2)\right)$, $t\in(0,T)$, with $T=21\ell^2/8$ so that $T = 42 > 4^2 = \ell^2$, thus violating the contraction condition \ref{['T-L2']}. Nevertheless, an excellent illustration of the decomposition \ref{['fi-hl-dec-i']} is still observed, as the discrepancy $dc(x,t)=q(x,t)-v(x,t)-w(x,t)$ is of $\mathcal{O}(10^{-3})$.

Theorems & Definitions (14)

  • Theorem 1: Finite interval to half-line I
  • Theorem 2: Finite interval to half-line II
  • Remark 2.1
  • Theorem 3: Nonlinear reaction-diffusion on a finite interval
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.2
  • Remark 2.3
  • ...and 4 more