Short-Time Plasma Evolution: Flow Generation and Magnetogenesis

We develop a self-consistent analytical two-fluid framework for plasma evolution in the short-time regime, elucidating the fundamental mechanism underlying the coupled generation of flow and magnetic fields. We show that consistency between ion momentum and mass conservation imposes a structural constraint on the system: the total pressure must satisfy the Laplace equation, $\nabla^2 P = 0$. This constraint enables a class of exact analytical solutions in which pressure gradients simultaneously drive plasma flow and generate magnetic fields through a Biermann-type mechanism. Using representative parameters, we obtain magnetic-field strengths and flow velocities consistent with both laser-produced plasmas and large-scale astrophysical systems. This framework provides a unified description of pressure-driven magnetogenesis and plasma flow in the short-time regime.