Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) the form $$\mathrm{d} X= u(\omega,t,X)\, \mathrm{d} t + \frac12 \sigma(\omega,t,X)\sigma'(\omega,t,X)\,\mathrm{d} \sigma(\omega,t,X) \, \mathrm{d}W(t), $$ where drift coefficient $u$ is random and irregular. The regular noise $\sigma$ may vanish. main contribution pathwise uniqueness under assumptions that belongs to $L^p(\Omega; L^\infty([0,T];\dot{H}^1(\mathbb{R})))$ any finite $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ as $t\downarrow 0$, satisfies one-sided gradient bound $\partial_x u(\omega,t,x) \le K(\omega, t)$, process $K(\omega,t )>0$ exhibits an exponential moment $\mathbb{E} \exp\Big(p\int_t^T K(s)\,\mathrm{d} s\Big) \lesssim {t^{-2p}}$ small times $t$, some $p\ge1$. This study motivated by ongoing work on Hunter--Saxton equation, perturbation nonlinear transport equation arises in modelling director field nematic liquid crystal. In this context, acts selection principle dissipative weak partial (SPDE).