\(L_p\)-Estimates for solutions to a hyperbolic system (Q2713923)

The author examines the following system of differential equations which appears in studying the dynamics of a rotating compressible fluid: NEWLINE\[NEWLINE \begin{aligned} &D_tu_1 + \alpha u_2 + D_{x_1}u_4 = 0, \\ &D_tu_2 - \alpha u_1 + D_{x_2}u_4 = 0, \\ &D_tu_3 + D_{x_3}u_4 = 0, \\ &D_tu_4 + D_{x_1}u_1 + D_{x_2}u_2 + D_{x_3}u_3 = 0. \end{aligned} NEWLINE\]NEWLINE As a matter of fact, this system is the so-called symmetric hyperbolic system and can be rewritten as NEWLINE\[NEWLINE D_tu + \sum_{j=1}^3A_jD_{x_j}u + Bu = 0, NEWLINE\]NEWLINE where \(A_j\) are symmetric matrices. As a result, the author proves the existence and uniqueness of solutions to the Cauchy problem in the space \(W^1_{p,\gamma}(\mathbb R^4_+)\) for \(p\neq 2\). The author discusses the effect of smoothness loss for generalized solutions and then the cases ensuring regularity of every \(L_p\)-bounded solution.