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Consider $$u_t-\alpha u_{xx} - \beta u_{xxx} = f(u_x)$$ with initial condition $u(0,x) = u_0(x)$,

where $\alpha>0$, $\beta \in \mathbb{R}$, $u_0 \in C^\infty(\mathbb{R})$, and $f$ is Lipschitz (that is, $|f(u_x) - f(v_x)| \le \Vert u_x - v_x \Vert_\infty$).

Where can I find or how can I prove a global wellposedness result (possibly of classical solutions) for this kind of equation?

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    $\begingroup$ Your equation is basically the Korteweg-de Vries-Burgers equation, so you could try searching there. $\endgroup$
    – user254433
    Jun 23, 2017 at 1:13
  • $\begingroup$ @user254433 Do you have any specific references? Also, I don't think that the KdV-Burgers should have classical solutions (because of the $uu_x$ term). $\endgroup$
    – Jun
    Jun 23, 2017 at 20:03
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    $\begingroup$ I have no specific Kdv-Burgers references, so I would recommend searching MathSciNet or Google Scholar if no one else replies. But Burgers $u_t+uu_x=u_{xx}$ has global classical solutions (the Hopf-Cole transformation actually yields a closed form solution). Moreover, by the inverse scattering transform, KdV $u_t+uu_x=-u_{xxx}$ is globally solvable. So I suspect your equation, which is a combination of these two, is also solvable. $\endgroup$
    – user254433
    Jun 24, 2017 at 6:33

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For the sake of simplicity, let me pick $f(u_x)=\frac{1}{2}u_x^2$ and $u_0\in H^5$. We have that, in the new variable $f=u_x$, the equation reads $$ f_t-\alpha f_{xx}-\beta f_{xxx}=ff_x $$

We have that $$ \frac{1}{2}\frac{d}{dt}\|f\|_{L^2}^2+\alpha\|f_x\|_{L^2}^2=0, $$ so $$ \|f(t)\|_{L^2}^2+2\alpha\int_0^t\|f_{x}\|_{L^2}^2ds\leq \|f(0)\|_{L^2}^2. $$ If we test the equation against $-f_{xx}$, then $$ \frac{1}{2}\frac{d}{dt}\|f_x\|_{L^2}^2+\alpha\|f_{xx}\|_{L^2}^2\leq \frac{1}{2\alpha}\|ff_x\|_{L^2}^2+\frac{\alpha}{2}\|f_{xx}\|_{L^2}^2. $$ Using that $$ \|ff_x\|_{L^2}^2\leq \|f\|_{L^4}^2\|f_x\|_{L^4}^2\leq C\|f\|_{H^{0.25}}^2\|f\|_{H^{1.25}}^2\leq C\|f\|_{L^2}^{3/2}\|f_{x}\|_{L^2}^{1/2}\|f_x\|_{L^2}^{3/2}\|f_{xx}\|_{L^2}^{1/2}, $$ thus, $$ \|ff_x\|_{L^2}^2\leq C\|f_{x}\|_{L^2}^{4}+\frac{\alpha}{2}\|f_{xx}\|_{L^2}^{1/2}. $$ Using Gronwall and the finiteness of $\int_0^t\|f_{x}\|_{L^2}^2ds$, we conclude. In this way, you can get a priori estimates for any Sobolev norm. After that you can just mollify your equation appropriately and pass to the limit in the approximate problems.

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  • $\begingroup$ Thank you. I'm interesting in learning in more detail what you mean by "mollify the equation appropriately". $\endgroup$
    – Jun
    Jul 6, 2017 at 21:51
  • $\begingroup$ For instance, you can check the book "Vorticity and Incompressible Flow" by Andrew J. Majda, Andrea L. Bertozzi (chapter 3 if I remember correctly). Basically the idea is that if you mollify, you can think on the problem as having an ODE in a Banach space (because you do not lose any derivative, you will put the derivative on the mollifier). $\endgroup$
    – guacho
    Jul 7, 2017 at 12:56

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