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Pure Mathematics 2024
双曲方程组Goursat问题整体光滑解
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Abstract:
本文主要研究一个广义压力下二维退化双曲守恒方程组Goursat问题存在解且具有光滑性。考虑该问题为混合型方程的Goursat问题，首先引入该方程组的特征角，并且对该特征角进行限制，求出特征角的方向导数从而得到压力P的特征分解。通过特征分解得到解的不变三角形，由此得出解的边界值估计以及局部存在性。根据压力P的特征分解，利用连续性方法建立解的梯度估计，将局部解扩展到全局，从而证明半双曲片整体解的存在性。
In this paper, we study the existence of global smooth solutions for Goursat problem of two- dimensional degenerated hyperbolic conservation systems under generalized pressure. Considering this problem as a Goursat problem with mixed equations, the characteristic angles of the equations are introduced first, and the directional derivatives of these characteristic angles are obtained by limiting the characteristic decomposition of pressure P that is obtained. The invariant triangle of the solutions is obtained by the method of characteristic decomposition, and the boundary value estimation and local existence of the solution are obtained. According to the characteristic decomposition of the pressure, the gradient estimation of the solution is established by the continuity method, and the local solutions are extended to the global solutions, so as to prove the existence of the global solutions of the semi-hyperbolic plates.
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