### EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME

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)

#### Abstract

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:
$$\mathbf{u}_t + (\mathbf{u}\nabla) \mathbf{u} = \varepsilon \triangle \mathbf{u}, \quad u_{\nu} (\mathbf{x}, -1, \varepsilon) = - x_{\nu} + 4^{-\nu}(\nu + 1) x_{\nu}^{2\nu + 1},$$
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:
$$\frac{\partial u_{\nu} (0, t, \varepsilon)}{\partial x_{\nu}} = \frac{1}{t} \left[ 1 + O \left( \varepsilon |t|^{- 1 - 1/\nu} \right) \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to -\infty, \quad t \to -0.$$
The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained:
$$u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty.$$

#### Keywords

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

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#### References

1. Arnold V.I. Singularities of Caustics and Wave Fronts. Math. Its Appl. Ser., vol. 62. Dordrecht: Kluwer Acad. Publ., 1990. 258 p. DOI: 10.1007/978-94-011-3330-2
2. Arnold V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of Differentiable Maps. Vol. I. Classification of Critical Points, Caustics and Wave Fronts. Monogr. Math., vol. 82. Boston, MA: Birkhäuser, 1985. 392 p.
3. Bogaevsky I.A. Reconstructions of singularities of minimum functions, and bifurcations of shock waves of the Burgers equation with vanishing viscosity St. Petersburg Math. J., 1990. Vol. 1, No. 4. P. 807–823.
4. Bogaevsky I.A. Perestroikas of shocks and singularities of minimum functions. Phys. D: Nonlinear Phenomena, 2002. Vol. 173, No. 1–2. P. 1–28. DOI: 10.1016/S0167-2789(02)00652-8
5. Gurbatov S.N., Rudenko O.V., Saichev A.I. Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics. Berlin: Springer, 2011. 472 p. DOI: 10.1007/978-3-642-23617-4
6. Gurbatov S.N., Saichev A.I., Shandarin S.F. Large-scale structure of the Universe. The Zeldovich approximation and the adhesion model. Phys. Usp., 2012. Vol. 55, No. 3. P. 223–249. DOI: 10.3367/UFNr.0182.201203a.0233
7. Il’in A.M. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Transl. Math. Monogr., vol. 102. Providence, R.I.: Am. Math. Soc., 1992. 281 p.
8. Zakharov S.V. Asymptotic solution of the multidimensional Burgers equation near a singularity. Theor. Math. Phys., 2018. Vol. 196, No. 1. P. 976–982. DOI: 10.1134/S0040577918070048
9. Zakharov S.V. Singular points and asymptotics in the singular Cauchy problem for the parabolic equation with a small parameter. Comp. Math. Math. Phys., 2020. Vol. 60, No. 5. P. 821–832. DOI: 10.1134/S0965542520050164