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

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:
$$
u_{\nu} (\mathbf{x}, t, \varepsilon) \approx
- 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu}
\tanh \left[ \frac{x_{\nu}}{\varepsilon}
\left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!,
\quad
\frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty.
$$