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On the forced Burgers equation with periodic boundary conditions. (English) Zbl 0930.35156

Giaquinta, M. (ed.) et al., Differential equations: La Pietra 1996. Conference on differential equations marking the 70th birthdays of Peter Lax and Louis Nirenberg, Villa La Pietra, Florence, Italy, July 3–7, 1996. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 65, 133-153 (1999).
This paper deals with the Burgers equation \[ u_t+ uu_x= \nu u_{xx}+ F_x(t,x), \] where \(F(t,x)\) is of unit period in both \(t\) and \(x\). The momentum \(\int_0^1u\) is a conserved quantity and thus can be prescribed by a given value \(c\). This equation has a unique \(\mathbb{Z}^2\)-periodic solution when \(\nu\neq 0\) and \(F\) is real analytic and moreover \(u\) is real analytic and monotone in \(c\). Some preliminary results are obtained when \(\nu\to 0\).
For the entire collection see [Zbl 0905.00061].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B10 Periodic solutions to PDEs
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