Boundedness of Solutions to a Degenerate Diffusion Equation

Abstract

The diffusion equation with a bounded saturation range under the time derivative and with Robin boundary conditions is shown to admit a regular bounded solution provided that the saturation function and the permeability coefficient have controlled decay at infinity. The result remains valid even if Preisach hysteresis is present in the pressure-saturation relation. The method of proof is based on a Moser-Alikakos iteration scheme which is compatible with a generalized Preisach energy dissipation mechanism.

Appendices

A Auxiliary Section A: Embeddings and Interpolations

We summarize here a few standard results from the theory of Lebesgue spaces L ^p and Sobolev spaces W ^k, p than all can be found, e. g., in the monograph [9]. Recall that we denote by | ⋅ | _p the norm in L ^p(Ω), and by ∥⋅ ∥ _p the norm in L ^p(Ω × (0, T)). We start with the interpolation inequality in Lebesgue spaces which we state in the following form.

Lemma A.1

Let 1 ≤ p ₁ < p < p ₀ ≤ ∞ be given. Then for every $v \in L^{p_{0}}(\varOmega )$ we have

$$\displaystyle{ \vert v\vert _{p} \leq \vert v\vert _{p_{0}}^{\eta }\vert v\vert _{p_{1}}^{1-\eta } }$$

(A.1)

with

$$\displaystyle{ \eta = \frac{ \frac{1} {p_{1}} -\frac{1} {p}} { \frac{1} {p_{1}} - \frac{1} {p_{0}} } \in (0,1). }$$

(A.2)

Indeed, the same formula holds if Ω is replaced with Ω × (0, T) and the norms | ⋅ | are replaced with ∥⋅ ∥.

We also refer to the Gagliardo-Nirenberg inequality:

Lemma A.2

There exists a constant C > 0 such that for every w ∈ W ^1,p(Ω) the inequality

$$\displaystyle{ \vert w\vert _{q} \leq C\left (\vert w\vert _{s} + \vert w\vert _{s}^{1-\rho }\vert \nabla w\vert _{ p}^{\rho }\right ) }$$

(A.3)

with

$$\displaystyle{\rho = \frac{\frac{1} {s} -\frac{1} {q}} {\frac{1} {s} + \frac{1} {N} -\frac{1} {p}} \in (0,1)}$$

is valid for all 1 ≤ s < q, 1∕q > (1∕p) − (1∕N),

as well as to the Poincaré inequality:

Lemma A.3

Under Hypothesis  2.1  (vii), there exists a constant C > 0 such that the inequality

$$\displaystyle{ \vert v\vert _{2}^{2} \leq C\left (\int _{\varOmega }\vert \nabla v\vert ^{2}\,\mathrm{d}x +\int _{ \partial \varOmega }b(x)\vert v\vert ^{2}\,\mathrm{d}s(x)\right ) }$$

(A.4)

holds for every v ∈ L ²(Ω).

We define the space

$$\displaystyle{X =\{ w \in W^{1,2}(\varOmega \times (0,T)): \nabla w \in L^{\infty }(0,T;L^{2}(\varOmega ))\},}$$

and denote by the symbol ↪ ↪ the compact embedding relation. The following anisotropic compact embeddings are established in [9]:

$$\displaystyle{ X\hookrightarrow \hookrightarrow L^{2}(\varOmega;C[0,T]), }$$

(A.5)

and

$$\displaystyle{ X\hookrightarrow \hookrightarrow L^{2}(\partial \varOmega \times (0,T)). }$$

(A.6)

B Auxiliary Section B: The Preisach Operator

We use here the variational definition of the Preisach operator which was shown in [19] to equivalent to the original construction in [23]. It is based on the variational inequality for the unknown function ξ _r

$$\displaystyle{ \left.\begin{array}{ll} \vert v(t) -\xi _{r}(t)\vert \leq r &\forall t \in [0,T]\,, \\ \dot{\xi }_{r}(t)(v(t) -\xi _{r}(t) - y) \geq 0\ \ \mbox{ a. e. }\ &\forall \vert y\vert \leq r\,, \\ \xi _{r}(0) = P_{r}(v(0))\,, \end{array} \right \} }$$

(B.1)

where r > 0 is a fixed constraint, P _r is the mapping defined by (3.2), v ∈ W ^1,1(0, T) is a given input, and the dot denotes time derivative. The mapping $\mathfrak{p}_{r}: W^{1,1}(0,T) \rightarrow W^{1,1}(0,T)$ which with v associates the solution ξ _r ∈ W ^1,1(0, T) of (B.1) is called the play operator according to [18]. It is proved in [18, §2] that for arbitrary two inputs v ₁, v ₂ ∈ W ^1,1(0, T), the inequality

$$\displaystyle{ \vert \mathfrak{p}_{r}[v_{1}](t) - \mathfrak{p}_{r}[v_{2}](t)\vert \leq \max _{\tau \in [0,t]}\vert v_{1}(\tau ) - v_{2}(\tau )\vert }$$

(B.2)

holds for all t ∈ [0, T], hence the play operator can be extended into a Lipschitz continuous mapping $\mathfrak{p}_{r}: C[0,T] \rightarrow C[0,T]$. Furthermore, directly from the definition (B.1) we can infer that the identity

$$\displaystyle{ \dot{\xi }_{r}(t)\dot{v}(t) = (\dot{\xi }_{r}(t))^{2} }$$

(B.3)

holds a. e. for every v ∈ W ^1,1(0, T). For inputs v depending on x ∈ Ω and t ∈ (0, T) we define the play operator $\mathfrak{p}_{r}: L^{p}(\varOmega;C[0,T]) \rightarrow L^{p}(\varOmega;C[0,T])$ for 1 ≤ p ≤ ∞ by the formula

$$\displaystyle{ \mathfrak{p}_{r}[v](x,t) = \mathfrak{p}_{r}[v(x,\cdot ](t) }$$

(B.4)

and by virtue of (B.2), the play is Lipschitz continuous in L ^p(Ω; C[0, T]) for all 1 ≤ p ≤ ∞.

Given a nonnegative function $\psi \in L^{\infty }(\varOmega;L^{1}((0,\infty ) \times \mathbb{R}))$, the Preisach operator G: L ^p(Ω; C[0, T]) → L ^p(Ω; C[0, T]) is for (x, t) ∈ Ω× [0, T] defined by the formula

$$\displaystyle{ G[v](x,t) =\int _{ 0}^{\infty }\int _{ 0}^{\mathfrak{p}_{r}[v](x,t)}\psi (x,r,z)\,\mathrm{d}z\,\mathrm{d}r\,, }$$

(B.5)

and ψ is called the Preisach density of G. If we denote for $(x,r,v) \in \varOmega \times (0,\infty ) \times \mathbb{R}$

$$\displaystyle{ \varPsi (x,r,v) =\int _{ 0}^{v}\psi (x,r,z)\,\mathrm{d}z\,, }$$

(B.6)

then (B.5) can be written in the form

$$\displaystyle{ G[v](x,t) =\int _{ 0}^{\infty }\varPsi (x,r,\mathfrak{p}_{ r}[v](x,t))\,\mathrm{d}r\,. }$$

(B.7)

The following statement is an easy consequence of (B.2).

Proposition B.1

G: L ^p(Ω; C[0, T]) → L ^p(Ω; C[0, T]) is Lipschitz continuous for every 1 ≤ p ≤ ∞.

From (B.3) it follows that for each v ∈ L ²(Ω; W ^1,1(0, T)) we have

$$\displaystyle{ (G[v])_{t}v_{t} \geq 0\quad \mbox{ a. e.} }$$

(B.8)

The Preisach operator is monotone in the sense of Hilpert’s inequality

$$\displaystyle{ (G[v_{1}] - G[v_{2}])_{t}H(v_{1} - v_{2}) \geq \frac{\partial } {\partial t}\int _{0}^{\infty }(\varPsi (x,r,\mathfrak{p}_{ r}[v_{1}]) -\varPsi (x,r,\mathfrak{p}_{r}[v_{2}]))^{+}\,\mathrm{d}r }$$

(B.9)

established in [17], which holds a. e. for all v ₁, v ₂ ∈ L ²(Ω; W ^1,1(0, T)) and where H is the Heaviside function (3.24) and (⋅ )⁺ denotes the positive part. A different proof can be found in [21, Proposition II.2.12].

Let $\lambda: \mathbb{R} \rightarrow \mathbb{R}$ be a nondecreasing function, λ(0) = 0, and let v ∈ L ²(Ω; W ^1,1(0, T)) be given. Put $\xi _{r} = \mathfrak{p}_{r}[v]$. By definition (B.1) of the play we have

$$\displaystyle{(\xi _{r})_{t}(v -\xi _{r}) \geq 0\quad \mbox{ a. e.},}$$

hence,

$$\displaystyle{(\xi _{r})_{t}(\lambda (v) -\lambda (\xi _{r})) \geq 0\quad \mbox{ a. e.}}$$

It follows that

$$\displaystyle{(G[v])_{t}\lambda (v) =\int _{ 0}^{\infty }\psi (x,r,\xi _{ r})(\xi _{r})_{t}\lambda (v)\,\mathrm{d}r \geq \int _{0}^{\infty }\psi (x,r,\xi _{ r})(\xi _{r})_{t}\lambda (\xi _{r})\,\mathrm{d}r\quad \mbox{ a. e.}}$$

and we conclude that

$$\displaystyle{ (G[v])_{t}\lambda (v) \geq (U_{\lambda }[v])_{t}\quad \mbox{ a. e.}, }$$

(B.10)

where

$$\displaystyle{ U_{\lambda }[v] =\int _{ 0}^{\infty }\int _{ 0}^{\mathfrak{p}_{r}[v]}\psi (x,r,z)\lambda (z)\,\mathrm{d}z\,\mathrm{d}r\,. }$$

(B.11)

This can be interpreted as a generalized hysteresis energy inequality with hysteresis potential U _λ , see [21, Chapter II].

Let us cite also the following result of [20].

Proposition B.2

Let $J:\varOmega \times \mathbb{R} \rightarrow \mathbb{R}: (x,v)\mapsto J(x,v)$ be a function such that $\frac{\partial J} {\partial v} \in L^{\infty }(\varOmega \times \mathbb{R})$ is positive almost everywhere, J(x, 0) = 0, J(x, ±∞) = ±∞ a. e., and let G be a Preisach operator with Preisach density $\psi \in L^{\infty }(\varOmega;L^{1}((0,\infty ) \times \mathbb{R}))$ , ψ(x, r, v) ≥ 0 a. e. Then the operator G _J defined by the formula

$$\displaystyle{ G_{J}[u](x,t) = G[J(x,u(x,\cdot ))](t) }$$

(B.12)

is a Preisach operator of the form  (B.5) with density

$$\displaystyle\begin{array}{rcl} & & \psi _{J}(x,r,z) = \frac{\partial J} {\partial v}\!\left (x, \frac{z+r} {2} \right )\frac{\partial J} {\partial v}\!\left (x, \frac{z-r} {2} \right ) \\ & & \quad \times \psi \left (x,J\!\left (x, \frac{z+r} {2} \right )-J\!\left (x, \frac{z-r} {2} \right ),J\!\left (x, \frac{z+r} {2} \right )+J\!\left (x, \frac{z-r} {2} \right )\right ).\qquad \quad {}\end{array}$$

(B.13)

Note that we have ψ _J ≥ 0 and

$$\displaystyle{ \int _{0}^{\infty }\int _{ -\infty }^{\infty }\psi _{ J}(x,r,z)\,\mathrm{d}z\,\mathrm{d}r =\int _{ 0}^{\infty }\int _{ -\infty }^{\infty }\psi (x,r,z)\,\mathrm{d}z\,\mathrm{d}r\quad \mbox{ a. e.} }$$

(B.14)

Acknowledgements

This research was supported by the GAČR Grant GA15-12227S and RVO: 67985840.