American Institute of Mathematical Sciences

$ iu_{t}+u_{x x}=-uv, $

$ v_{t t}-v_{x x}+v=|u|^2,$

$u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$

is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.

$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad $ ($*$)

$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow \pm\infty.$

Assume that the corresponding Riemann problem

$u_t+F(u)_x=0,$

$ u(0,x)=u^r_0(x)=u_-,\quad x<0, and u_+,\quad x>0$

can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is a small perturbation of an approximate rarefaction wave constructed in Section 2, then we show that the Cauchy problem ($*$) admits a unique global smooth solution $u(t,x)$ which tends to $ u^r(t,x)$ as the $t$ tends to infinity. Here, we do not require $|u_+ - u_-|$ to be small and thus show the convergence of the corresponding global smooth solutions to strong rarefaction waves for $2\times 2$ viscous conservation laws.

$ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy.

We establish regularity, radial symmetry, and monotonicity of the solutions. We also consider subcritical cases, super critical cases, and singular solutions in all cases; and obtain qualitative properties for these solutions.