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- W119217348 abstract "The Burgers’ equation $$partial _tu+upartial _xu=mupartial_x^2u,;;;;;;;;;;;(1.1)$$ t < 0,x ∈ R, u = u(t,x),u(0,x) = u 0(x), admits the well-known Hopf-Cole explicit solution $$u(t,x)=frac{int _{-infty }^infty[(x-y)/t]exp[(2mu)^{-1}(xi(y)-(x-y)^2/2t)]dy}{int _{-infty }^infty exp[(2mu)^{-1}(xi(y)-(x-y)^2/2t)]dy};;;;;;;;;;(1.2)$$ where $$xi (x)=-int_{x}^{-infty } u_0(y)dy$$ (see Hopf (1950)). It describes propagation of nonlinear hyperbolic waves, and has been considered as a model equation for hydrodynamic turbulence (see e.g. Chorin (1975)). Due to nonlinearity, the solution (1.2) enters several different stages, including that of shock waves’ formation, which are largely determined by the value of the Reynolds number R = σl/μ (see Gurbatov, Malakhov, Saichev (1991)). Here, μ < 0 is the viscosity parameter, while σ and l have the physical meaning of characteristic scale and amplitude of ξ(x), respectively." @default.
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- W119217348 date "1994-01-01" @default.
- W119217348 modified "2023-10-16" @default.
- W119217348 title "Burgers’ Topology on Random Point Measures" @default.
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- W119217348 doi "https://doi.org/10.1007/978-1-4612-0253-0_13" @default.
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