Burgers' type equation with vanishing higher order doi:10.3934/cpaa.2007.6.505
Naoki Fujino - Graduate School of Pure and Applied Sciences, University of Tsukuba, 305-8571, Ibaraki, Japan (email) Abstract: We consider a scalar conservation law of Burgers' type: $u_t+(u^2/2)_x = \varepsilon u_{x x}-\delta u_{x x x}+\gamma u_{x x x x x}$ ($(x, t)\in \mathbf R \times$ (0, ∞)). We prove that if $\varepsilon$, $\delta=\delta(\varepsilon)$, $\gamma=\gamma(\varepsilon)$ tend to $0$, then for $q\in (2, 16/5)$, the sequence {$u^\varepsilon$} of solutions converges in $L^k(0, T^\star; L^p(\mathbf R)) (k< $ ∞, p$<$q) to the unique entropy solution $u\in L^\infty (0, T^\star; L^q(\mathbf R))$ to the inviscid Burgers equation $u_t+(u^2/2)_x = 0$. More precisely we show that, under the condition $\delta=O(\varepsilon^{3/(3-q)})$ and $\gamma=O(\varepsilon^4$ $\delta^{(8q-7)/9})$ for $q\in(2,3)$ or $\delta=O(\varepsilon^{12/(19-4q)}$ $\gamma^{3/(19-4q)})$ and $\gamma=O(\varepsilon^{4}$ $\delta^{(8q-7)/9})$ for $q\in[3,16/5)$, the limit of the sequence is the entropy solution. Moreover if we assume the uniform boundedness of {$u^\varepsilon(\cdot,t)$} in $L^q(\mathbf R)$ for $q>2$, the condition $\delta=o(\varepsilon^3)$ and $\gamma=o(\varepsilon^4\delta)$ is sufficient to establish the conclusion. We derive new a priori estimates which enable to use the technique of the compensated compactness, the Young measures and the entropy measure-valued solutions.
Keywords: Scalar conservation laws, Burgers equation, entropy
solutions, Young measures, measure-valued solutions.
Received: May 2006; Revised: December 2006; Published: March 2007. |
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