EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME
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...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2022-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/439 |
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| author | Sergey V. Zakharov |
| author_facet | Sergey V. Zakharov |
| author_sort | Sergey V. Zakharov |
| collection | DOAJ |
| description | 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:
$$
u_{\nu} (\mathbf{x}, t, \varepsilon) \approx
- 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu}
\tanh \left[ \frac{x_{\nu}}{\varepsilon}
\left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!,
\quad
\frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty.
$$ |
| format | Article |
| id | doaj-art-c4ecc5e4031f4e76a4812baa7d3d73da |
| institution | DOAJ |
| issn | 2414-3952 |
| language | English |
| publishDate | 2022-07-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-c4ecc5e4031f4e76a4812baa7d3d73da2025-08-20T02:51:41ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522022-07-018110.15826/umj.2022.1.012141EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIMESergey V. Zakharov0Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, 620108The 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: $$ u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty. $$https://umjuran.ru/index.php/umj/article/view/439vector burgers equation, cauchy problem, cole–hopf transform, singular point, laplace's method, multiscale asymptotics. |
| spellingShingle | Sergey V. Zakharov EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME Ural Mathematical Journal vector burgers equation, cauchy problem, cole–hopf transform, singular point, laplace's method, multiscale asymptotics. |
| title | EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME |
| title_full | EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME |
| title_fullStr | EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME |
| title_full_unstemmed | EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME |
| title_short | EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME |
| title_sort | evolution of a multiscale singularity of the solution of the burgers equation in the 4 dimensional space time |
| topic | vector burgers equation, cauchy problem, cole–hopf transform, singular point, laplace's method, multiscale asymptotics. |
| url | https://umjuran.ru/index.php/umj/article/view/439 |
| work_keys_str_mv | AT sergeyvzakharov evolutionofamultiscalesingularityofthesolutionoftheburgersequationinthe4dimensionalspacetime |