Geodesic
Exponential stability of the flow for~a~generalized Burgers equation on~a~circle
Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 588-598

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The paper deals with the problem of stability for the flow of the $\mathrm{1D}$ Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on $\mathbb{R}$, which is exponentially stable in $H^1$ as $t\to+\infty$. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability $1$.
Keywords: Burgers equation, exponential stability, bounded trajectory. 
@article{CMFD_2023_69_4_a2,
     author = {A. Djurdjevac and A. R. Shirikyan},
     title = {Exponential stability of the flow for~a~generalized {Burgers} equation on~a~circle},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {588--598},
     publisher = {mathdoc},
     volume = {69},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2023_69_4_a2/}
}
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A. Djurdjevac; A. R. Shirikyan. Exponential stability of the flow for~a~generalized Burgers equation on~a~circle. Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 588-598. http://geodesic.mathdoc.fr/item/CMFD_2023_69_4_a2/