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Two Classes of Nonlocal Evolution Equations Related by a Shared Traveling Wave Problem

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From Particle Systems to Partial Differential Equations (PSPDE 2015)

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Abstract

We consider nonlocal reaction-diffusion equations and nonlocal Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations.

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Acknowledgements

The author was partially supported by Austrian Science Fund (FWF) under grant P28661 and the FWF-funded SFB #F65.

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Appendices

Appendix A: Caputo Fractional Derivative on \(\pmb {\mathbb {R}}\)

For \(\alpha >0\), the (Gerasimov–)Caputo derivatives are defined as, see [42, 54],

$$\begin{aligned} (\mathscr {D}^\alpha _{+} f)(x)&= {\left\{ \begin{array}{ll} f^{(n)}(x) &{} \text {if } \alpha =n\in \mathbb {N}_0 \ , \\ \tfrac{1}{\varGamma (n-\alpha )}\int _{-\infty }^{x} \; { \frac{f^{(n)}(y)}{(x-y)^{\alpha -n+1}} } \; \,{\text {d}}\!y\, &{} \text {if } n-1<\alpha<n \text { for some } n\in \mathbb {N}_0 \ . \end{array}\right. } \\ (\mathscr {D}^\alpha _{-} f)(x)&= {\left\{ \begin{array}{ll} f^{(n)}(x) &{} \text {if } \alpha =n\in \mathbb {N}_0 \ , \\ \tfrac{(-1)^n}{\varGamma (n-\alpha )}\int _{x}^{\infty } \; { \frac{f^{(n)}(y)}{(y-x)^{\alpha -n+1}} } \; \,{\text {d}}\!y\, &{} \text {if } n-1<\alpha <n \text { for some } n\in \mathbb {N}_0 \ . \end{array}\right. } \end{aligned}$$

Properties:

  • For \(\alpha >0\) and \(\lambda > 0\)

    $$\begin{aligned} (\mathscr {D}^\alpha _{+} \exp (\lambda \cdot ))(x) = \lambda ^\alpha \exp (\lambda x) \ , \quad (\mathscr {D}^\alpha _{-} \exp (-\lambda \cdot ))(x) = \lambda ^\alpha \exp (-\lambda x) \end{aligned}$$
  • For \(\alpha >0\) and \(f\in \mathscr {S}(\mathbb {R})\), a Caputo derivative is a Fourier multiplier operator with \((\mathscr {F}\mathscr {D}^\alpha _{+} f)(k) = ({\text {i}}k)^\alpha (\mathscr {F}f)(k)\) where \(({\text {i}}k)^\alpha = \exp (\alpha \pi {\text {i}}{{\mathrm{sgn}}}(k)/2)\).

  • If \(\bar{u}\) is the profile of a TWS \((\bar{u},c)\) in the sense of Definition 1, then

    $$\begin{aligned} \int _{-\infty }^{\infty } \; { \bar{u}'(y)\, \mathscr {D}^\alpha _{+} \bar{u}(y) } \; \,{\text {d}}\!y\, = \tfrac{1}{2} \int _{\mathbb {R}} \; { \bar{u}'(x) \int _{\mathbb {R}} \; { \frac{\bar{u}'(y)}{|x-y|^\alpha } } \; \,{\text {d}}\!y\, } \; \,{\text {d}}\!x\, \ge 0 \ , \end{aligned}$$
    (A.1)

    where the last inequality follows from [46, Theorem 9.8].

Appendix B: Shock Wave Theory for Scalar Conservation Laws

A standard reference on the theory of conservation laws is [29], whereas [45] covers the special topic of non-classical shock solutions. A scalar conservation law is a partial differential equation

$$\begin{aligned} \partial _{t} {u} + \partial _{x} {f(u)} = 0 \ , \quad t>0 \ , \quad x\in \mathbb {R}\ , \end{aligned}$$
(B.1)

for some flux function \(f:\mathbb {R}\rightarrow \mathbb {R}\). For nonlinear functions f, it is well known that the initial value problem (IVP) for (B.1) with smooth initial data may not have a classical solution for all time \(t>0\) (due to shock formation). However, weak solutions may not be unique. The Riemann problems are a subclass of IVPs for (B.1), and especially important in some numerical algorithms: For given \(u_-, u_+\in \mathbb {R}\), find a weak solution u(xt) for the initial value problem of (B.1) with initial condition

$$\begin{aligned} u(x,0) = {\left\{ \begin{array}{ll} u_-\ , &{} x<0 \ , \\ u_+\ , &{} x>0 \,. \end{array}\right. } \end{aligned}$$
(B.2)

Weak solutions of a Riemann problem that are discontinuous for \(t>0\) may not be unique.

Example 1

A shock wave is a discontinuous solution of the Riemann problem,

$$\begin{aligned} u(x,t) = {\left\{ \begin{array}{ll} u_-\ , &{} x<ct \ , \\ u_+\ , &{} x>ct \ , \end{array}\right. } \end{aligned}$$
(B.3)

if the shock triple \((u_-,u_+;c)\) satisfies the Rankine–Hugoniot condition

$$\begin{aligned} f(u_+)-f(u_-) = c(u_+-u_-) \,. \end{aligned}$$
(B.4)

The Rankine–Hugoniot condition (B.4) is a necessary condition that \(u_\pm \) are stationary states of an associated TWE (28), see (30).

1.1 Shock Admissibility

Classical approaches to select a unique weak solution of the Riemann problem are

  1. (a)

    Lax’ entropy condition:

    $$\begin{aligned} f'(u_+)<c<f'(u_-) \ . \end{aligned}$$
    (B.5)

    It ensures that in the method of characteristics all characteristics enter the shock/discontinuity of a shock solution (B.3). For convex flux function f, condition (B.5) reduces to \(u_->u_+\). Shocks satisfying (B.5) are also called Lax or classical shocks. For non-convex flux functions f, also non-classical shocks can arise in experiments, called slow undercompressive shocks if \(f'(u_\pm )>c\), and fast undercompressive shocks if \(f'(u_\pm )<c\).

  2. (b)

    Oleinik’s entropy condition.

    $$\begin{aligned} \frac{f(w)-f(u_-)}{w-u_-} \ge \frac{f(u_+)-f(u_-)}{u_+-u_-}\, \text {for all}\, w\, \text {between}\, u_-\, \text {and}\, u_+. \end{aligned}$$
    (B.6)
  3. (c)

    Entropy solutions satisfying integral inequalities based on entropy-entropy flux pairs, such as Kruzkov’s family of entropy-entropy flux pairs.

  4. (d)

    Vanishing viscosity. In the classical vanishing viscosity approach, instead of (B.1) one considers for \(\varepsilon >0\) equation

    $$\begin{aligned} \partial _{t} {u} + \partial _{x} {f(u)} = \varepsilon \partial ^{2}_{x} {u} \ , \quad t>0 \ , \quad x\in \mathbb {R}\ , \end{aligned}$$
    (B.7)

    where \(\varepsilon \partial ^{2}_{x} {u}\) models diffusive effects such as friction. Equation (B.7) is a parabolic equation, hence the Cauchy problem has global smooth solutions \(u^\varepsilon \) for positive times, especially for Riemann data (B.2). An admissible weak solution of the Riemann problem is identified by studying the limit of \(u^\varepsilon \) as \(\varepsilon \searrow 0\).

    In other applications, different higher order effects may be important. For example, a nonlocal generalized KdVB equation (1) can be interpreted as a scalar conservation law (B.1) with higher-order effects \(\mathscr {R}[u] := \varepsilon \mathscr {L}_1[u] +\delta \partial _{x} {} \mathscr {L}_2[u]\).

    Already for convex functions f, the convergence of solutions of the regularized equations (e.g. (1)) to solutions of (B.1) reveals a diverse solution structure. The solutions of viscous conservation laws (B.7) converge for \(\varepsilon \searrow 0\) to Kruzkov entropy solutions of (B.1). In contrast, in case of KdVB equation (4) with \(f(u) = u^{2}\) the limit \(\varepsilon ,\delta \rightarrow 0\) depends on the relative strength of diffusion and dispersion:

    • Weak dispersion: \(\delta =O(\varepsilon ^2)\) for \(\varepsilon \rightarrow 0\) e.g. \(\delta =\beta \varepsilon ^2\) for some \(\beta >0\).

      TWS converge strongly to entropy solution of Burgers equation.

    • Moderate dispersion: \(\delta =o(\varepsilon )\) for \(\varepsilon \rightarrow 0\) includes weak dispersion.

      TWS converge strongly to entropy solution of Burgers equation, see [51].

    • Strong dispersion: weak limit of TWS for \(\varepsilon ,\delta \rightarrow 0\) may not be a weak solution of Burgers equation.

    For non-convex flux functions f, a TWS may converge to a weak solution of (B.1) which is not an Kruzkov entropy solution, but a non-classical shock.

A simplistic shock admissibility criterion based on the vanishing viscosity approach is the existence of TWS for a given shock triple:

Definition 6

(compare with [41]) A solution u of the Riemann problem is called admissible (with respect to a fixed regularization \(\mathscr {R}\)), if there exists a TWS \((\bar{u},c)\) in the sense of Definition 1 of the regularized equation (e.g. (1)) for every shock wave with shock triple \((u_-,u_+;c)\) in the solution u.

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Achleitner, F. (2017). Two Classes of Nonlocal Evolution Equations Related by a Shared Traveling Wave Problem. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations. PSPDE 2015. Springer Proceedings in Mathematics & Statistics, vol 209. Springer, Cham. https://doi.org/10.1007/978-3-319-66839-0_2

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