Analysis and Applied Mathematics Seminar

Bingyu Zhang.
U. of Cincinnati
On Some Joint Effects of Dispersion and Dissipation of a Class of Nonlinear Evolution Equations
Abstract: It is known that the solutions of the Cauchy problem for the Korteweg-de Vries (KdV) equation $ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in T, \ t\in R,$
and the viscous Burgers equation
$ u_t +uu_x - u_{xx} =0, \quad u(x,0)= \phi (x), \quad x\in T, \ t>0 $
posed on a periodic domain $T$, do not possess the sharp Kato smoothing property:
$ \phi \in H^s (T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (T, L^2 (0,T))$.
Here, we discuss the equation,
$ u_t +uu_x +\alpha (x,t) u_{xxx} - \beta (x,t)u_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in T, \ t\geq 0, $
and demonstrate that if
$\int _{\mathbb{T}}\frac{\beta (x,t)}{|\alpha (x,t)|} dx >0 \quad \forall t\geq 0,$
and if it is locally well-posed in the space $ H^s (T)$ with $s \geq 0$, then its solution $u$ possesses the sharp Kato smoothing property,
$ \phi \in H^s (T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (T, L^2 (0,T)), \quad \forall \, s\geq 0. $
In addition, the nonlinear part of its solution $u$ possesses the strong Kato smoothing property, $ \phi \in H^s (T) \implies (u -v)\in C([0,T]; H^{s+1} (T)), \quad \forall \, s>\frac12, $
and the double sharp Kato smoothing property $ \phi \in H^s (T) \implies \partial ^{s+2}_x(u -v)\in L^{\infty}_x (\T, L^2 (0,T)), \quad \forall \, s>\frac12, $
with $v$ being the solution of the linear problem
$ v_t+ \alpha (x,t)v_{xxx} - \beta (x,t) v_{xx} =0, \quad v(x,0)=\phi (x), \quad x\in T, \ t>0. $
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