J. Math. Fluid Mech. 20 (2018), 189–197
2017 Springer International Publishing
Journal of Mathematical
Existence and Stability of Spatial Plane Waves for the Incompressible Navier–Stokes
Sim˜ao Correia and M´ario Figueira
Communicated by H. Beir˜ao da Veiga
Abstract. We consider the three-dimensional incompressible Navier–Stokes equation on the whole space. We observe that
this system admits a L
family of global spatial plane wave solutions, which are connected with the two-dimensional
equation. We then proceed to prove local well-posedness over a space which includes L
) and these solutions. Finally,
we prove L
-stability of spatial plane waves, with no condition on their size.
Mathematics Subject Classiﬁcation. 35B35, 35Q30, 76D03.
Keywords. Incompressible Navier–Stokes, local well-posedness, stability, spatial plane waves.
In this work, we consider the Cauchy problem for the incompressible Navier–Stokes equation on R
− Δu +(u ·∇)u = ∇p, t > 0,u(t,·):R
div u =0
u(0) = u
We shall focus on the d = 3 case. We seek to study spatial plane waves, that is,
u(t, x, y, z)=g(t, x − cz, y),p(t, x, y, z)=q(t, x − cz, y),u
(x, y, z)=g
(x − cz, y),c∈ R. (1.1)
We shall refer to c as the speed of the wave and g,q as the wave proﬁles. The idea of considering such
solutions ﬁrst appeared in  in the context of the hyperbolic nonlinear Schr¨odinger equation. Since the
existence of such solutions is quite trivial in such a framework, the attention was then directed to the
local well-posedness over a space which includes H
functions and spatial plane waves. Finally, it was
proven, in some cases, that H
perturbations of spatial plane waves are stable. These ideas were later
developed for the nonlinear Schr¨odinger equation (see ), where one considers superpositions of waves
with diﬀerent speeds (either a numerable collection or a continuous one).
The generality of such results made us search for other models where one could try to apply these
ideas, such as the (NS). A considerable change in the framework is observed: on one hand, the change
from a dispersive equation to a diﬀusive one; on the other, the passage from a scalar equation to a system.
Moreover, there is an intrinsic interest in obtaining existence and stability results for the three-dimensional
(NS) (and so this work is not simply an academic problem).
The existence of a class of global spatial plane waves is proven by observing that the proﬁle satisﬁes
a two-dimensional (NS) system (cf. Proposition 4). Naturally, these solutions will not belong to L
for any p<∞. However, under some regularity assumptions over the initial data g
, they will belong to
). We then derive a local well-posedness result over a space which includes these global solutions and
) (cf. Theorem 5). Finally, we prove the stability of spatial plane waves under L
without any smallness condition of the proﬁle of the wave. As a consequence, our result proves that, if