一类p-守恒律组的双曲性和真非线性条件、弱解和熵条件
Hyperbolicity and genuine nonlinearity conditions for certain p-systems of conservation laws, weak solutions and the entropy condition
摘要：
我们考虑一维弹性理论中出现的p-守恒律组。这种系统由一个称为应变能函数的函数$W$确定。我们考虑文献中已知的四种形式的$W$。这些是圣维南-基尔霍夫，奥格登，基尔霍夫修改，布拉茨-科-奥格登形式。在每一种情况下，我们确定参数$ρ_0$、$μ$和$λ$的条件，在这些条件下，相应的系统是双曲的和真正的非线性的。我们还建立了初边值问题边值问题的弱解的定义.接下来，我们集中在一个特定的问题，其弱解是通过达朗贝尔公式在线性理论中得到的。在所考虑的情况下，p-系统是非线性的，所以我们解决他们采用Rankine-Hugoniot条件。最后，我们问，如果这样的解决方案满足熵条件。对于一个标准的熵函数，我们提供了一个完整的答案，除了Blatz-Ko-Ogden的情况。对于一般的严格凸熵函数，在双曲性和真正的非线性假设下，结果是当速度函数的初值接近于零时，这些解满足熵条件。
英文摘要：
We consider a p-system of conservation laws that emerges in one dimensional elasticity theory. Such system is determined by a function $W$, called strain-energy function. We consider four forms of $W$ which are known in the literature. These are St.Venant-Kirchhoff, Ogden, Kirchhoff modified, Blatz-Ko-Ogden forms. In each of those cases we determine the conditions for the parameters $ρ_0$, $μ$ and $λ$, under which the corresponding system is hyperbolic and genuinely nonlinear. We also establish what it means a weak solution of an initial and boundary value problem. Next we concentrate on a particular problem whose weak solution is obtained in a linear theory by means of D'Alembert's formula. In cases under consideration the p-systems are nonlinear, so we solve them employing Rankine-Hugoniot conditions. Finally we ask if such solutions satisfy the entropy condition. For a standard entropy function we provide a complete answer, except of the Blatz-Ko-Ogden case. For a general strictly convex entropy function the result is that for the initial value of velocity function near zero these solutions satisfy the entropy condition, under the assumption of hyperbolicity and genuine nonlinearity.
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