Transport equations with inflow boundary conditions

Abstract

We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late Charles Amick, we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions.

Spaces

Here we collect the notation used for various Sobolev spaces and norms. We denote by $L^p(\Omega )$ the Lebesgue spaces [7] of p-th power integrable functions, with norm

$$\begin{aligned} \Vert \, f \,\Vert _{L^p(\Omega )}=\bigg (\int _{\Omega } |f({{\mathbf {x}}})|^p \, dx\bigg )^{1/p}. \end{aligned}$$

Note that we can easily apply the same notation to vector or tensor valued f. We think of tensors of any arity as vectors of the appropriate length, and we think of $|f({{\mathbf {x}}})|$ as the Euclidean length of this vector. For tensors of arity 2 (i.e., matrices) this is the same as the Frobenius norm. We will write the spaces for such tensor-valued functions as $L^p(\Omega )^m$ for the appropriate m (e.g., $m=d^2$ for arity 2). Similarly, we denote by $L^\infty (\Omega )$ the Lebesgue space of essentially bounded functions, with

$$\begin{aligned} \Vert \, f \,\Vert _{L^\infty (\Omega )}=\sup \left\{ |f({{\mathbf {x}}})| \; \big | \; \hbox {a.e.}\;{{\mathbf {x}}}\in \Omega \right\} . \end{aligned}$$

Correspondingly, we define Sobolev spaces and norms of order m by

$$\begin{aligned} \Vert \, f \,\Vert _{W^m_p(\Omega )}=\bigg (\sum _{|\alpha |\le m}\Vert \, D^\alpha f \,\Vert _{L^p(\Omega )}^p\bigg )^{1/p}, \end{aligned}$$

where $D^\alpha$ is the weak derivative $\partial ^\alpha /\partial {{\mathbf {x}}}^{|\alpha |}$ [7]. More precisely, the spaces $W^m_p(\Omega )$ are defined as the subspaces of $L^p(\Omega )$ for which the corresponding norm is finite. The case $p=2$ is denoted by H:

$$\begin{aligned} H^m(\Omega )=W^m_2(\Omega ). \end{aligned}$$

Correspondingly, we define

$$\begin{aligned} \Vert \, \nabla ^m f \,\Vert _{L^p(\Omega )}= \bigg (\sum _{|\alpha |= m}\Vert \, D^\alpha f \,\Vert _{L^p(\Omega )}^p\bigg )^{1/p}. \end{aligned}$$

Note that

$$\begin{aligned} \Vert \, f \,\Vert _{W^1_p(\Omega )}\le \Vert \, f \,\Vert _{L^p(\Omega )}+\Vert \, \nabla f \,\Vert _{L^p(\Omega )} \le 2^{(p-1)/p)} \Vert \, f \,\Vert _{W^1_p(\Omega )}. \end{aligned}$$

(69)

We will briefly use the space $H^1_0(\Omega )$ of $f\in H^1(\Omega )$ such that $f=0$ on $\partial \Omega$. The dual space $H^{-1}(\Omega )^d$ is the set of Schwartz distributions [24] for which the dual norm

$$\begin{aligned} \Vert \, {{\mathbf {u}}} \,\Vert _{H^{-1}(\Omega )}=\sup _{\mathbf{0}\ne {\varvec{\phi }}\in H^1_0(\Omega )^d} \frac{\langle {{\mathbf {u}}}\cdot {\varvec{\phi }}\rangle }{\Vert \, {\varvec{\phi }} \,\Vert _{H^{1}(\Omega )}} \end{aligned}$$

is finite.