let's consider the general Burgers' equation

$$ \frac{\partial u}{\partial t} + c(x) \frac{\partial u}{\partial x} = \nu \frac{\partial ^{2}u}{\partial x^{2}} $$

where $c(x)$ is a periodic and bounded function and $v$ is a constant.

I want to state sufficient conditions such that the problem is well-posed. I know that a problem is well-posed if there exist a solution $u(x,t)$ for the initial condition $g(x)$ s.t $$ ||u(\cdot,t)||_{L_2} \leq k e^{\alpha t} ||g(\cdot)||_{L_2}$$ Here I arbitrary chose the $L_2$ norm.

Given the definition, how can I state sufficient conditions on $c(x)$ and $v$ such that the Burgers' equation is well-posed?



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