Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. (English) Zbl 0842.35076

We derive a detailed, pointwise description of the asymptotic behavior of solutions of the Cauchy problem for the Navier-Stokes equations of compressible flow in several space dimensions, with initial data in \(L^1 \cap H^{k (n)}\). We show that, asymptotically, the solution decomposes into the sum of two terms, one of which dominates in \(L^p\) for \(p > 2\), the other for \(p < 2\). The dominant term for \(p > 2\) has constant density and divergence-free momentum field, decaying at the rate of a heat kernel. Thus, as measured in \(L^p\) for \(p > 2\), all smooth, small-amplitude solutions of the Navier-Stokes equations are asymptotically incompressible. When \(p < 2\), the dominant term reflects instead the spreading effect of convection, and decays more slowly than a heat kernel; in fact, the solution may grow without bound in \(L^p\) for \(p\) near 1.
These features of the solution do not arise in one-dimensional flow, nor are they apparent from previously known \(L^2\) decay rates. An alternative interpretation of our estimates shows that the solution is asymptotically given by a diffusion about the origin whose mass is that of the initial data, convected by the fundamental solution of the linearized Euler equations. This result thus defines the correct notion of “diffusion wave” in the context of compressible, Navier-Stokes flows. Our analysis involves a number of interesting issues in Fourier multiplier and Paley-Wiener theory, required for the derivation of pointwise bounds from representations of the Fourier transforms of Green’s matrices.


35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B40 Asymptotic behavior of solutions to PDEs
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