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The kinetic theory for the instabilities driven by the Hall current with asheared current velocity, which has the method of the shearing modes or theso-called non-modal approach as its foundation, is developed. The developedtheory predicts that in the Hall plasma with the inhomogeneous electric field,the separate spatial Fourier mode of the perturbations is determined in theframe convected with one of the plasma components. Because of the differentshearing of the ion and electron flows in the Hall plasma, this mode isperceived by the second component as the Doppler-shifted continuously shearedmode with time-dependent wave numbers. Due to this effect, the interaction ofthe plasma components forms the nonmodal time-dependent process, which shouldbe investigated as the initial value problem. The developed approach is appliedto the solutions of the linear initial value problems for the hydrodynamicmodified two-stream instability and the kinetic ion-sound instability of theplasma with a sheared Hall current with a uniform velocity shear. Thesesolutions reveal that the uniform part of the current velocity is responsiblefor the modal evolution of the instability, whereas the current velocity shearis the source of the development of the nonmodal instability with exponentgrowing with time as $\sim\left(t-t_0\right)^3$.

The nonmodal kinetic theory for the electrostatic instabilities of a plasma with a sheared Hall current