Sergey V. Zakharov     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, 620108, Russian Federation)


The solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation \(\varepsilon\) in the \(4\)-dimensional space-time is studied:
\mathbf{u}_t + (\mathbf{u}\nabla) \mathbf{u} =
\varepsilon \triangle \mathbf{u},
u_{\nu} (\mathbf{x}, -1, \varepsilon) =
- x_{\nu} + 4^{-\nu}(\nu + 1) x_{\nu}^{2\nu + 1},
With the help of the Cole–Hopf transform \(\mathbf{u} = - 2 \varepsilon \nabla \ln H,\) the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field \(\mathbf{u}\) on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established:
\frac{\partial u_{\nu} (0, t, \varepsilon)}{\partial x_{\nu}}
= \frac{1}{t} \left[ 1 + O \left( \varepsilon |t|^{- 1 - 1/\nu} \right) \right]\!,
\frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to -\infty,
t \to -0.
The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained:
u_{\nu} (\mathbf{x}, t, \varepsilon) \approx
- 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu}
\tanh \left[ \frac{x_{\nu}}{\varepsilon}
\left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!,
\frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty.


Vector Burgers equation, Cauchy problem, Cole–Hopf transform, Singular point, Laplace's method, Multiscale asymptotics.

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DOI: http://dx.doi.org/10.15826/umj.2022.1.012

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