The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation
The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation
复制标题
来自真实消失粘度近似的边界黎曼求解器
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
L. Spinolo
中科院分区：
文献类型：
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作者：
S. Bianchini;L. Spinolo
摘要我们研究了双曲线-抛物线逼近的极限
$$Left{egin{数组}{L@{quad}L@{quad}L}v^{varepsilon}_t+ilde{A}
Left(v^{varepsilon}，，varepsilon v^{varepsilon}_x
Ight)v^{varepsilon}_x=varepsilon ilde{B}(v^{varepsilon})v^{varepsilon}_{xx}
Mathbb{R}^N cr ilde中的quad v^{varepsilon}
M ss}(v^{varepsilon}(t，，0))等价ar g cr v^{varepsilon}(0，，x)等价ar{v}_0。结束{数组}
$$函数$${ilde{ss}}$$被定义为即使$${ilde{B}}$$不可逆，也保证初边值问题是良好适定的。数据$${ar{g}}$$和$${ar{v}_{0}}$$是常量。当$${ilde{B}}$$可逆时，前面的问题采用更简单的形式$$Left{egin{数组}{L@{quad}L@{quad}L}v^{varepsilon}_t+ilde{A}Left(v^{varepsilon}，，varepsilon v^{varepsilon}_x
Ight)v^{varepsilon}_x=varepsilon ilde{B}(v^{varepsilon})v^{varepsilon}_{xx}quad v^{varepsilon}in mathbb{R}^N cr v^{varepsilon}(t，，0)等价ar v_b cr v^{varepsilon}(0，，x)等价ar{v}_0。结束{数组}
同样，数据$${ar{v}_b}$$和$${ar{v}_0}$$是常量。在以前的公式中包含了保守的情况。假设了$${v^{varepsilon}}$$的收敛、总变差小等技术假设，并给出了极限的完整刻画。最有趣的地方如下：首先，考虑边界特征情况，即$${ilde{A}}$$的一个特征值可以为0。其次，正如前面指出的，我们考虑了$${ilde{B}}$$不可逆的可能性。为了处理这种情况，我们采用川岛和静田提出的条件作为假设条件，依靠物理上有意义的例子。我们还引入了块线性退化的一个新条件。我们证明，如果不满足这一条件，则可能发生病理行为。
AbstractWe study the limit of the hyperbolic–parabolic approximation
$$left{egin{array}{l@{quad}l@{quad}l} v^{varepsilon}_t + ilde{A}
left(v^{varepsilon}, , varepsilon v^{varepsilon}_x
ight) v^{varepsilon}_x = varepsilon ilde{B}(v^{varepsilon} ) v^{varepsilon}_{xx}
quad v^{varepsilon} in mathbb{R}^N cr ilde{
m ss}(v^{varepsilon} (t, , 0)) equiv ar g cr v^{varepsilon} (0, , x) equiv ar{v}_0. end{array}
ight.$$The function $${ ilde {ss}}$$ is defined in such a way as to guarantee that the initial boundary value problem is well posed even if $${ ilde {B}}$$ is not invertible. The data $${ar {g}}$$ and $${ar {v}_{0}}$$ are constant. When $${ ilde {B}}$$ is invertible, the previous problem takes the simpler form$$left{egin{array}{l@{quad}l@{quad}l} v^{varepsilon}_t + ilde{A}left(v^{varepsilon}, , varepsilon v^{varepsilon}_x
ight) v^{varepsilon}_x = varepsilon ilde{B}(v^{varepsilon} ) v^{varepsilon}_{xx}quad v^{varepsilon} in mathbb{R}^N cr v^{varepsilon} (t, , 0) equiv ar v_b cr v^{varepsilon} (0, , x) equiv ar{v}_0. end{array}
ight.$$Again, the data $${ar {v}_b}$$ and $${ar {v}_0}$$ are constant. The conservative case is included in the previous formulations. Convergence of the $${v^{varepsilon}}$$ , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of $${ ilde {A}}$$ can be 0. Second, as pointed out before, we take into account the possibility that $${ ilde {B}}$$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if this condition is not satisfied, then pathological behaviors may occur.