Particular solutions to one-dimensional Cauchy problems for scalar parabolic-hyperbolic conservation laws and their applications
Particular solutions to one-dimensional Cauchy problems for scalar parabolic-hyperbolic conservation laws and their applications
复制标题
标量抛物双曲守恒定律一维柯西问题的特解及其应用
DOI：
10.1007/s00030-022-00775-2
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Watanabe Hiroshi
中科院分区：
文献类型：
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作者：
Atsushi Atsuji;Hiroshi Kaneko;牧野 哲;Tetsutaro Shibata;Watanabe Hiroshi
在本文中，我们引入多重间断行波解到一维抛物-双曲守恒律Cauchy问题。由于方程中含有非线性对流项和退化扩散项，因此它既具有双曲型方程的性质，又具有抛物型方程的性质。因此，很难研究（CP）的解的行为。克服困难的方法之一是构造特殊的解并研究它们的性质。在纯双曲情形下，由于黎曼解是自相似解，所以研究得比较多。然而，这在本文中是不能期望的。因此，我们专注于行波结构，而不是自相似结构。实际上，我们在扩散项的某些限制下成功地构造了至多为1-间断的激波型行波。本文去掉扩散项的限制，构造了（CP）的多重间断激波型行波，并研究了它们的性质。接下来，我们讨论所构造的行波的稳定性。第三，我们构造了（CP）的稀疏波型上、下解，并证明了它们的稳定性。最后，我们估计的传播速度的支持熵解决方案（CP）使用构造的功能。
In this paper, we introduce traveling waves with multiple discontinuities to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equation has nonlinear convective term and degenerate diffusion term, it has both properties of hyperbolic equations and those of parabolic equations. Therefore, it is difficult to investigate the behavior of solutions to (CP). One way to overcome difficulties is to construct particular solutions and investigate their properties. In pure hyperbolic case, Riemann solutions are well studied because they are self-similar solutions. However, it cannot be expected in this paper. Hence, we focus on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with at most 1-discontinuity under some restriction of the diffusion term. In the present paper, we remove the restriction of the diffusion term and construct shock wave type traveling waves with multiple discontinuities to (CP) and investigate their properties. Next, we discuss the stability of the constructed traveling waves. Thirdly, we construct rarefaction wave type sub-, super-solutions to (CP) and prove the stability of them. Finally, we estimate the propagation speed of support for entropy solutions to (CP) using the constructed functions.