[Paper Review] A singular limit problem for the Kudryashov-Sinelshchikov equation

Giuseppe Maria Coclite, Lorenzo di Ruvo|arXiv (Cornell University)|Nov 18, 2014

Nonlinear Waves and Solitons6 references3 citations

TL;DR

This paper investigates the singular limit of the Kudryashov-Sinelshchikov equation as diffusion and dispersion parameters vanish, proving that solutions converge strongly in $L^p_{ ext{loc}}$ to the unique entropy solution of the Burgers equation under the condition $\beta = \mathcal{O}(\varepsilon^4)$. The analysis relies on a priori estimates and the compensated compactness method in $L^p$ spaces.

ABSTRACT

We consider the Kudryashov-Sinelshchikov equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation coverge to the entropy ones of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of the compansated compactness method in the L^p setting.

Motivation & Objective

• To analyze the singular limit of the Kudryashov-Sinelshchikov equation as the diffusion and dispersion parameters tend to zero.
• To establish the convergence of solutions of the dispersive-diffusive equation to the entropy solution of the inviscid Burgers equation.
• To justify the validity of the Burgers equation as a zero-dispersion-diffusion limit of the more complex Kudryashov-Sinelshchikov model.
• To provide rigorous $L^p$-based compactness arguments for nonlinear PDEs with third-order nonlinear dispersion and dissipation.

Proposed method

• Derives a priori estimates for the Kudryashov-Sinelshchikov equation in $L^2$ and $L^4$ norms under initial data in $L^2 \cap L^4$.
• Applies the compensated compactness method in the $L^p$ setting to handle weak convergence and nonlinear terms.
• Imposes the scaling condition $\beta = \mathcal{O}(\varepsilon^4)$ to control higher-order derivatives and ensure convergence.
• Uses smooth approximations of initial data satisfying uniform bounds in $L^2$, $L^4$, and $H^1$ norms.
• Employs Hölder and Cauchy-Schwarz inequalities to estimate error terms involving third-order derivatives.
• Establishes strong convergence in $L^p_{\text{loc}}$ for $1 \leq p < 4$ via energy and compactness arguments.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of the Kudryashov-Sinelshchikov equation converge to the entropy solution of the Burgers equation as $\varepsilon, \beta \to 0$?
• RQ2 How does the relative scaling of $\beta$ and $\varepsilon$ affect the convergence of solutions in the singular limit?
• RQ3 Can the compensated compactness method be applied to third-order nonlinear dispersive-diffusive equations with mixed nonlinearities?
• RQ4 What role do $L^p$ estimates play in proving strong convergence for such singular limits?
• RQ5 Is the entropy solution of the Burgers equation the unique limit of solutions to the Kudryashov-Sinelshchikov equation under the given scaling?

Key findings

• Solutions of the Kudryashov-Sinelshchikov equation converge strongly in $L^p_{\text{loc}}(\mathbb{R}^+ \times \mathbb{R})$ for $1 \leq p < 4$ as $\varepsilon, \beta \to 0$ under the condition $\beta = \mathcal{O}(\varepsilon^4)$.
• The limit function $u$ is the unique entropy solution of the inviscid Burgers equation $\partial_t u + A u \partial_x u = 0$.
• The convergence is established via uniform bounds on initial data and energy estimates in $L^2$ and $L^4$ norms.
• The compensated compactness method successfully handles the nonlinear third-order terms $\partial_x(u \partial_{xx}^2 u)$ and $\partial_x u \partial_{xx}^2 u$ in the limit process.
• The proof relies on the decay of higher-order derivative terms, shown via Hölder and Cauchy-Schwarz estimates under the $\mathcal{O}(\varepsilon^4)$ scaling.
• The result holds not only for the standard parameter choice but also extends to cases where $A = (C + \alpha)^{2n}$ under a suitable constraint on $\alpha$.