I am struggling to understand the behavior of the Fourier transform (in the $x$ variable) of initially smooth solutions of the hyperbolic Burgers' equation in 1-D,
$ \partial_t u + u~ \partial_x u =0$ .
I start with a smooth and rapidly decaying initial condition $u(x,t)=u_0(x)$ on $\Bbb R$ . This solution evolves in time until it breaks down. At the time of first breakdown $t=T$ I look at the Fourier transform $\hat u(k,T)$ of the solution $u(x,T)$.
In particular, I am trying hard to understand how and why the $L^p$ norms of the Fourier transform $\hat u$ remain finite at the time of first blow-up for $p>1$. I think that if one uses weak (or Lorentz) norms, then this non-blow-up extends even to the weak $L^1$ norm.
The only way I have been able to understand this property is via the convervation law for the $L^\infty$ norm of $u$. For the $\|u \|_{L^\infty} $ norm to be defined at the time of first blow-up, the Fourier transform needs to remain in a weak $L^1$ space. Interpolation explains the rest.
My question is whether there is a way to understand the non-blow-up of the said $L^p$ norms of the Fourier transform $\hat u$ without invoking the conservation law for the $L^\infty$ norm of $u$.
What I seek is some kind of direct Fourier-analytic way to see what is going on. I have reached an impasse.
I will be very grateful for any insight or advice.