Energy conservation and Onsager's conjecture for a surface growth model

In this paper, it is shown that the energy equality of weak solution v to a surface growth model is valid if v(x) epsilon L-p(0, T; L-q(T)) with 3/p + 1/q = 1 and 1 <= q <= 4, or v epsilon L-infinity( 0, T; L-infinity( T)), or v(xx) epsilon L-p( 0, T; L-q( T)) with 2/p + 2/5q = 1 and q >= 1, which gives an affirmative answer to a question proposed by Yang in [28, J. Differential Equations 283: 71-84, 2021]. Furthermore, Onsager's conjecture for this model is also considered.