Analogue of the Cole-Hopf transform for the incompressible Navier-Stokes equations and its application
Publisher: Taylor & Francis
arxiv: Mathematics::Analysis of PDEs | Physics::Fluid Dynamics
We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navi... View more
 J.D. Cole, On a linear quasilinear parabolic equation in aerodynamics, Q. Appl. Math. 9 (1951), pp. 225-236.
 E. Hopf, The partial differential equation ut + uux = μuxx, Commun. Pure Appl. Math. 3 (1950), pp. 201-230.
 F. Gesztesy and H. Holden, The Cole-Hopf and Miura transformations revisited, in Mathematical physics and stochastic analysis: essays in honour of Ludwig Streit, p.198, (ed.) S. Albeverio, P. Blanchard, L. Ferreira, T. Hida, Y. Kondratiev, R. Vilela Mendes, World Scientific, 2000, Singapore.
 A. Biryuk, Note on the transformation that reduces the Burgers equation to the heat equation, No. MP-ARC-2003-370 (2003).
 K. Ohkitani, A miscellany of basic issues on incompressible fluid equations, Nonlinearity 21 (2008), pp. 255-271.
 C. Nore, A. Malek and M.E. Brachet, Decaying Kolmogorov turbulence in a model of superflow, Phys. Fluids (1997), pp. 2644-2669.
 J.M. Steele, Stochastic Calculus and Financial Applications, Springer, New York, 2001.
 M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer, New York, 1996.
 P. Constantin, A.J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), 1495-1533.
 K. Ohkitani, Dynamical equations for the vector potential and the velocity potential in incompressible irrottational Euler flows: a refined Bernoulli theorem, Phys. Rev. E. 92 (2015), 033010.