Differential and Integral Equations

Fractional derivatives and smoothing in nonlinear conservation laws

Gustaf Gripenberg and Stig-Olof Londen

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It is shown that the solution of the Riemann problem $$ \hbox{$ \frac {\partial}{\partial t}$} \int_0^t k(t-s)(u(s,x)-u_0(x))\,\rm{d}s + (\sigma(u))_x(t,x) = 0, $$ where $u_0 = \chi_{\rminus}$ is continuous when $t> 0$. Here $k$ is locally integrable, nonnegative, and nonincreasing on $\mathbb{R}plus$ with $k(0+)=\infty$.

Article information

Differential Integral Equations, Volume 8, Number 8 (1995), 1961-1976.

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L65: Conservation laws
Secondary: 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35] 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]


Gripenberg, Gustaf; Londen, Stig-Olof. Fractional derivatives and smoothing in nonlinear conservation laws. Differential Integral Equations 8 (1995), no. 8, 1961--1976. https://projecteuclid.org/euclid.die/1369056135

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