Shock waves for radiative hyperbolic-elliptic systems
Shock waves for radiative hyperbolic-elliptic systems
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DOI：
10.1512/iumj.2007.56.3043
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发表时间：
2006-06
影响因子：
1.1
通讯作者：
Corrado Lattanzio;C. Mascia;D. Serre
中科院分区：
文献类型：
--
作者：
Corrado Lattanzio;C. Mascia;D. Serre
本文研究了双曲椭圆耦合系统ut + f（u）x + Lqx = 0，{ x ∈ R，t > 0，-qxx + Rq + G ·ux = 0，其中u ∈ Rn，q ∈ R且R > 0，G，L ∈ Rn.流函数f：Rn → Rn是光滑的，使得对于任意u，Vf有n个不同的真实的特征值.考虑了可容许辐射激波的存在性问题，即，存在形式为（u，q）（x，t）：=（U，Q）（x-st）的解，使得（U，Q）（±∞）=（u±，0），且u ± ∈ Rn，s ∈ R定义了约化双曲方程组的激波，通过形式上把L = 0得到.本文证明了，如果u_（？）λ k（u_）·rk（u_）<$0（其中λ k表示k阶本征值，rk是相应的右本征向量），（lk（u_）·L）（G ·rk（u_））> 0，则存在u_的邻域U，使得对任意u + ∈ U，s ∈ R，（u_，u + ; s）定义了约化双曲方程组的激波，则完备双曲-椭圆方程组存在（唯一的直到移位）容许的辐射激波.证明是基于减少系统的情况下，标量的情况下，一般严格凸通量的存在性问题被认为是，推广现有的结果Burgers'通量f（u）= u2/2。此外，我们能够证明，当冲击的大小时，轮廓（U，Q）获得平滑性|u + - u_|是足够小的，如先前证明的Burgers'通量的情况下。最后，非凸通量的一般情况下也处理，显示出类似的结果的存在性和规律性的配置文件。
The present paper deals with the following hyperbolicelliptic coupled system, modelling dynamics of a gas in presence of radiation, u t + f(u) x + Lq x = 0, { x ∈ R, t > 0, -q xx + Rq + G · u x = 0, where u ∈ R n , q ∈ R and R > 0, G, L ∈ R n . The flux function f: R n → R n is smooth and such that Vf has n distinct real eigenvalues for any u. The problem of existence of admissible radiative shock wave is considered, i.e., existence of a solution of the form (u,q)(x,t):= (U,Q)(x - st), such that (U,Q)(±∞) = (u±, 0), and u ± ∈ R n , s ∈ R define a shock wave for the reduced hyperbolic system, obtained by formally putting L = 0. It is proved that, if u_ is such that ∇λ k (u_)·r k (u_) ≠ 0 (where λ k denotes the k-th eigenvalue of ∇ f and r k a corresponding right eigenvector), and (l k (u_) ·L) (G ·r k (u_)) > 0, then there exists a neighborhood U of u_ such that for any u + ∈ U, s ∈ R such that the triple (u_, u + ; s) defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic-elliptic system. The proof is based on reducing the system case to the scalar case, hence the problem of existence for the scalar case with general strictly convex fluxes is considered, generalizing existing results for the Burgers' flux f(u) = u 2 /2. Additionally, we are able to prove that the profile (U, Q) gains smoothness when the size of the shock |u + - u_ | is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.