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On the \(L^2\)-instability of fluid flows. (English) Zbl 1062.35055

The author considers the incompressible Euler equations in two dimensions in the \((x_1,x_2)\)-plane in the strip \(|x_2|< 1\), for a steady basic flow having the form \(u_0(x)= (U(x_2), 0)\), i.e. a parallel flow having velocity profile \(U(x_2)\). The main result states that for every smooth profile \(U(x_2)\neq\text{const}\), the flow \(u_0(x)\) is unstable in \(L_2\). This means that there exists \(C> 0\), s.t. for every \(\varepsilon> 0\) there exists \(T> 0\) and a smooth solution \(v(x,t)\) of the Euler equations, such that \(\| v(x,0)- u_0(x)\|_{L^2}< \varepsilon\), but \(\| v(x,T)- u_0(x)\|_{L^2}> C\).

MSC:

35Q05 Euler-Poisson-Darboux equations
76E05 Parallel shear flows in hydrodynamic stability
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
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