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Multidimensional potential Burgers turbulence. (English) Zbl 1338.60153
Commun. Math. Phys. 342, No. 2, 441-489 (2016); erratum ibid. 344, No. 1, 369-370 (2016).
A multidimensional generalized stochastic Burgers equation \[ \mathbf u_t+(\nabla f(\mathbf u)\cdot\nabla)\mathbf u-\nu\Delta\mathbf u=\nabla\dot w \] is considered on the torus \(\mathbb T^d\), where the solution \(\mathbf u\) is sought in the form \(\mathbf u=\nabla\psi\), where \(\psi\) satisfies \[ d\psi+f(\nabla\psi)\,dt-\nu\Delta\psi\,dt=dw. \] Here, \(w\) is a Wiener process with a colored spatial covariance, \(f\) is a strongly convex function that satisfies a certain growth condition and \(\nu\) is a small positive number. The author proves a lot of estimates and asymptotics for the solutions \(\psi\) and \(\mathbf u\), e.g., estimates and asymptotics of various forms (upper, lower, maximal, integral) for Sobolev norms of \(\psi\) and \(\mathbf u\) as well as estimates and asymptotics for averaged increments (including directional and longitudinal) of \(\psi\) and \(\mathbf u\). Finally, the unique stationary measure for the latter equation above is studied and a rate of convergence to this measure is provided.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
76F55 Statistical turbulence modeling
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