Optimal bounds on the Kuramoto-Sivashinsky equation
Let \(u\) be a solution to the one-dimensional Kuramoto-Sivashinsky equation \[ \partial_t u + \partial_x \left( \frac{u^2}{2} \right) + \partial_x^2 u + \partial_x^4 u = 0 , \quad (t,x)\in (0,\infty)\times (0,L), \] with \(L\)-periodic boundary conditions and zero mean value. It is already known that \[ \limsup_{t\to \infty} \left( \frac{1}{L} \int_0^L u(t,x)^2 dx \right)^{1/2} \leq o(L), \] but conjectured that the \(o(L)\) in the above estimate could be improved to a constant. This bound is here improved (in a different topology) to \[ \left( \limsup_{T\to\infty} \frac{1}{TL} \int_0^T \int_0^L \left| |\partial_x|^\alpha u(t,x) \right|^p dxdt \right)^{1/p} \leq C (\ln{L})^\gamma, \] for any \(L\geq 2\), \(\gamma>5/3\), \(\alpha\in (1/3,2)\), and \(p\in (1,10/(3+\alpha))\), where \(|\partial_x|^\alpha u\) is the fractional gradient of \(u\) defined by its Fourier transform: \(\mathcal{F}(|\partial_x|^\alpha u)(q) = |q|^\alpha \mathcal{F}(u)(q)\). The proof relies on careful estimates at the level of the Littlewood-Paley decomposition of \(u\) and quantitative comparison with the entropy formulation of the Burgers equation with particular source terms. For the latter, an extension of Oleinik's one-sided gradient estimate is derived.
scientific article; zbMATH DE number 5602675