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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.

##### MSC:
 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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