# On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations
## 2 A Priori Estimates
We begin by observing that by (42), we have that  
\[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]4  
Multiplying (1) by \(-2\partial_{x}^{6}u\), thanks to (56), an integration on \(\mathbb{R}\) gives  
\[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]5 \[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]6 \[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]7 \[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]8 \[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq\int_{ \mathbb{R}}|\partial_{x}^{2}u||\partial_{x}^{4}u|dx\leq\frac{1}{2}\left\| \partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\left\| \partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}. \tag{47}\]9 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]0 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]1  
Therefore, we have  
\[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]2 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]3 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]4 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]5  
Due to (22), (29), (48), (49) and the Young inequality,  
\[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]6 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]7\[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]8 \[\left\|\partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T), \tag{48}\]9 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]0 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]1 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]2 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]3 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]4 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]5 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]6 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]7 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]8 \[\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\int_{0}^{t}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}ds\leq C(T), \tag{49}\]9 \[\partial_{x}^{3}\left(u^{3}\right)=6(\partial_{x}u)^{3}+18u\partial_{x}u \partial_{x}^{2}u+3u^{2}\partial_{x}^{3}u. \tag{50}\]0 \[\partial_{x}^{3}\left(u^{3}\right)=6(\partial_{x}u)^{3}+18u\partial_{x}u \partial_{x}^{2}u+3u^{2}\partial_{x}^{3}u. \tag{50}\]1 \[\partial_{x}^{3}\left(u^{3}\right)=6(\partial_{x}u)^{3}+18u\partial_{x}u \partial_{x}^{2}u+3u^{2}\partial_{x}^{3}u. \tag{50}\]2 \[\partial_{x}^{3}\left(u^{3}\right)=6(\partial_{x}u)^{3}+18u\partial_{x}u \partial_{x}^{2}u+3u^{2}\partial_{x}^{3}u. \tag{50}\]3  
where \(D_{2}\) is a positive constant which will specified later.