  A new $L^p$-primal-dual weak Galerkin method ($L^p$-PDWG) with $p>1$ is
proposed for the first-order transport problems. The existence and uniqueness
of the $L^p$-PDWG numerical solutions is established. In addition, the
$L^p$-PDWG method offers a numerical solution which retains mass conservation
locally on each element. An optimal order error estimate is established for the
primal variable. A series of numerical results are presented to verify the
efficiency and accuracy of the proposed $L^p$-PDWG scheme.
