OCCURRENCE AND NON-APPEARANCE OF SHOCKS IN FRACTAL BURGERS EQUATIONS
OCCURRENCE AND NON-APPEARANCE OF SHOCKS IN FRACTAL BURGERS EQUATIONS
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DOI：
10.1142/s0219891607001227
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发表时间：
2007-09
影响因子：
0.7
通讯作者：
Nathael Alibaud;J. Droniou;J. Vovelle
中科院分区：
文献类型：
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作者：
Nathael Alibaud;J. Droniou;J. Vovelle
我们认为分形汉堡方程(也就是说汉堡方程,增加了部分的拉普拉斯算子),我们证明,如果涉及到拉普拉斯算子的力量小于1/2,然后方程不调整初始条件:相反,如果拉普拉斯算子的力量大于1/2,不连续的初始数据可以在解决方案和冲击持续发展甚至对光滑的初始数据。我们还证明了只有在足够“大”的初始条件下才能产生激波，通过给出一个结果，该结果表明，对于光滑的“小”初始数据，解至少保持Lipschitz连续。
We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently "large" initial conditions, by giving a result which states that, for smooth "small" initial data, the solution remains at least Lipschitz continuous.