<p>The Stefan problem provides a fundamental model for phase-change phenomena, such as melting and solidification processes. While traditional formulations primarily focus on pure conduction, many scientific applications involve additional transport mechanisms, particularly advection driven by fluid flow. This study extends the classical Stefan problem by incorporating advection, offering a more comprehensive understanding of complex heat transfer dynamics. Using the unified transform method, also known as the Fokas method, we derive an existence theorem for the solution of the Stefan problem with advection. We also propose a method for determining boundary influx in this setting. Finally, we establish a double logarithmic stability estimate for the boundary flux condition, demonstrating that the inverse Stefan problem with advection can be severely ill-posed.</p>