Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability (Q2717292)

The author considers the Korteweg-de Vries-Burgers equation with an external excitation NEWLINE\[NEWLINEu_t+ u_x+ uu_x- u_{xx}+ u_{xxx}= f(x, t),\tag{1}NEWLINE\]NEWLINE \(x\in [0,1]\), \(t\geq 0\), with \(f(x,t)\) given.NEWLINENEWLINENEWLINEThe initial conditions are \(u(x,0)= \phi(x)\), and the boundary conditions are \(u(0,t)= u(1,t)= 0\). Two problems are posed:NEWLINENEWLINENEWLINE1) If the external excitation \(f(x,t)\) is periodic in time with period \(\omega\), then is the solution also time periodic and with the period \(\omega\)? If such a solution exists, it is easy to show that it is a limit cycle in the phase space of the initial-boundary value system (1).NEWLINENEWLINENEWLINE2) What kind of stability does this limit cycle process?NEWLINENEWLINENEWLINEThe author observes that the solution to this dynamic problem evolves in time according to a \(C_0\) semigroup \(W(t)\phi= u(t)\), with \(x\) suppressed. \(W(t)\) is generated by the operator \(A: A\psi= -\psi'''+ \psi'+ \psi''\), \(\psi\in H^3(0,1)\). The author now writes a formal solution \(u(t)\) as a convolution integral NEWLINE\[NEWLINEu(t)= \int^t_0 W(t- \tau)f(.,\tau) d\tau.NEWLINE\]NEWLINE Substituting \(v= u_t\) he locates \(v\) in a metric space \(Y^j_{\tau, T}\), which for \(T= 0\) is denoted by \(Y^j_{\tau}\), the superscript \(j\) is either \(0\) or \(3\). This is a metric space in which the map \(\Gamma\) representing the action of the semigroup on \(v\) is shown to be a contraction, implying the existence of a fixed point, which is the solution to the original system of equations. Several norm estimates precede this conclusion. Next, using several norm inequalities combined with the Gronwall inequality, the author proves that on an arbitrary interval \([t,t+ T]\) the norm of \(u\) is dominated by the \(L^2\) norm of the inhomogeneous term. It is shown in conclusion that for external excitation that is periodic and has small amplitude, the system has a unique solution, which is also time periodic with the same period, satisfying the prescribed boundary conditions. This periodic solution forms a limit cycle. A stability result, which the author proves next, implies that the set of such solutions forms an inertial manifold for the system.NEWLINENEWLINENEWLINEThis is an important result for this large class of equations with external forcing term.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00047].