Energy and Large Time Estimates for Nonlinear Porous Medium Flow with Nonlocal Pressure in $\mathbb {R}^N$

Abstract

We study the general family of nonlinear evolution equations of fractional diffusive type $\partial _t u-\text{ div }\left( |u|^{m_1}\nabla (-\Delta )^{-s} [ |u|^{m_2-1}u]\right) =f$. Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider $m_1, m_2>0$, and $s\in (0,1)$, and prove the existence of weak solutions. Moreover, when $f\equiv 0$ we obtain the $L^p$-$L^\infty$ decay estimates of solutions, for $p\geqq 1$. In addition, we also investigate the finite time extinction of solution.

Appendix

Lemma 10

Let $s\in (0,1)$. For any $\varepsilon >0$, we have

$$\begin{aligned} 0\leqq \mathcal {F} \left\{ \mathcal {L}^s_\varepsilon \right\} \leqq \mathcal {F} \left\{ (-\Delta )^s \right\} . \end{aligned}$$

Proof

It is known that for any $u\in \mathcal {S}(\mathbb {R}^N)$ (the Schwartz space), $\mathcal {F} \{(-\Delta )^s\}$ can be considered as a multiplier of $\mathcal {F} \{(-\Delta )^s u \}$, that is,

$$\begin{aligned} \mathcal {F}\left\{ (-\Delta )^s u \right\} (\xi ) = \mathcal {F} \{(-\Delta )^s\} \mathcal {F}\{u\} (\xi ). \end{aligned}$$

We have

$$\begin{aligned} (-\Delta )^s u(x) = \frac{1}{2} \int _{\mathbb {R}^N} \frac{u(x+h)+ u(x-h) -2 u(x) }{|h|^{N+2s}} \hbox {d}h. \end{aligned}$$

Taking the Fourier transform yields

$$\begin{aligned} \mathcal {F} \left\{ (-\Delta )^s u \right\} (\xi )&= \frac{1}{2} \int _{\mathbb {R}^N} \frac{e^{i\xi \cdot h}+ e^{-i\xi \cdot h} -2}{|h|^{N+2s}} \hbox {d}h \, \mathcal {F}\{u\}(\xi ) \\&= \int _{\mathbb {R}^N} \frac{1-\cos (\xi \cdot h)}{|h|^{N+2s}} \hbox {d}h \, \mathcal {F}\{u\} (\xi ) . \end{aligned}$$

This implies that

$$\begin{aligned} \mathcal {F} \left\{ (-\Delta )^s \right\} (\xi ) = \int _{\mathbb {R}^N} \frac{1-\cos (\xi \cdot h)}{|h|^{N+2s}} \hbox {d}h . \end{aligned}$$

(5.1)

Similarly, we also have

$$\begin{aligned} \mathcal {F} \left\{ \mathcal {L}^s_\varepsilon u \right\} (\xi )&= \frac{1}{2} \int _{\mathbb {R}^N} \frac{e^{i\xi \cdot h}+ e^{-i\xi \cdot h} -2}{(|h|^2 + \varepsilon ^2)^{\frac{N+2s}{2}}} \hbox {d}h \, \mathcal {F}\{u\}(\xi ) \\&= \int _{\mathbb {R}^N} \frac{1-\cos (\xi \cdot h)}{(|h|^2 + \varepsilon ^2)^{\frac{N+2s}{2}}} \hbox {d}h \, \mathcal {F}\{u\} (\xi ) . \end{aligned}$$

Therefore,

$$\begin{aligned} \mathcal {F} \left\{ \mathcal {L}^s_\varepsilon \right\} (\xi ) = \int _{\mathbb {R}^N} \frac{1-\cos (\xi \cdot h)}{(|h|^2 + \varepsilon ^2)^{\frac{N+2s}{2}}} \hbox {d}h . \end{aligned}$$

(5.2)

Then the conclusion of Lemma 10 follows from (5.1) and (5.2). $\quad \square$

Next, we have the following embedding results:

Lemma 11

Let $\alpha \in (0,1)$, $N\geqq 1$, and $p\geqq 1$. Then, we have

$$\begin{aligned} \Vert \nabla ^{\alpha } v \Vert _{L^p(\mathbb {R}^N)} \leqq C \Vert \nabla v \Vert ^{\alpha }_{L^p(\mathbb {R}^N)} \Vert v\Vert _{L^p(\mathbb {R}^N)}^{1-\alpha }, \quad \forall v\in W^{1,p}(\mathbb {R}^N). \end{aligned}$$

Proof

We have

$$\begin{aligned} \Vert \nabla ^{\alpha } v \Vert _{L^p(\mathbb {R}^N)}&\leqq C(N,\alpha ) \left( \int _{\mathbb {R}^N} \left( \int _{\mathbb {R}^N} \frac{|v(x+h)- v(x)|}{|h|^{N+\alpha }} \hbox {d}h \right) ^p \hbox {d}x \right) ^{1/p} \nonumber \\&\leqq C(N,\alpha , p)\left[ \left( \int _{\mathbb {R}^N} \left( \int _{|h|\leqq \lambda } \frac{|v(x+h)- v(x)|}{|h|^{N+\alpha }} \hbox {d}h \right) ^p \hbox {d}x \right) ^{1/p} \right. \nonumber \\&\quad \left. + \left( \int _{\mathbb {R}^N} \left( \int _{|h|> \lambda } \frac{|v(x+h)- v(x)|}{|h|^{N+\alpha }} \hbox {d}h \right) ^p \hbox {d}x \right) ^{1/p} \right] := C(\mathbf {I}_1 + \mathbf {I}_2) . \end{aligned}$$

(5.3)

Now we consider $\mathbf {I}_1$. Applying Young’s inequality and Hölder’s inequality yields

$$\begin{aligned} \mathbf {I}_1&\leqq \int _{ |h|\leqq \lambda } \left( \int _{\mathbb {R}^N} \left| \frac{|v(x+h)- v(x)|}{|h|^{N+\alpha }}\right| ^{p} \hbox {d}x \right) ^{1/p} \hbox {d}h \nonumber \\&\leqq \int _{|h|\leqq \lambda } \left( \int _{\mathbb {R}^N} \left( \int ^1_0 |\nabla v( t(x+h)+(1-t)x )| \hbox {d}t \right) ^{p} \hbox {d}x \right) ^{1/p} |h|^{-(N+\alpha -1)} \hbox {d}h \nonumber \\&\leqq \int _{|h|\leqq \lambda } \left( \int ^1_0 \int _{\mathbb {R}^N} |\nabla v( t(x+h)+(1-t)x )|^p \hbox {d}x\hbox {d}t \right) ^{1/p} |h|^{-(N+\alpha -1)} \hbox {d}h \nonumber \\&\leqq C(N, \alpha ) \lambda ^{1-\alpha } \Vert \nabla v\Vert _{L^p(\mathbb {R}^N)} . \end{aligned}$$

(5.4)

Next, we apply Young’s inequality to get

$$\begin{aligned} \mathbf {I}_2&\leqq \int _{|h|> \lambda } \left( \int _{\mathbb {R}^N} |v(x+h)- v(x)|^p \hbox {d}x \right) ^{1/p} |h|^{-(N+\alpha )} \hbox {d}h \nonumber \\&\leqq 2 \Vert v\Vert _{L^p(\mathbb {R}^N)} \int _{|h|> \lambda }|h|^{-(N+\alpha )} \hbox {d}h= C(N,\alpha ) \lambda ^{-\alpha } \Vert v\Vert _{L^p(\mathbb {R}^N)} . \end{aligned}$$

(5.5)

A combination of (5.4) and (5.5) implies

$$\begin{aligned} \mathbf {I}_1 + \mathbf {I}_2 \leqq C(N,\alpha ) \left( \lambda ^{1-\alpha } \Vert \nabla v\Vert _{L^p(\mathbb {R}^N)} + \lambda ^{-\alpha } \Vert v\Vert _{L^p(\mathbb {R}^N)} \right) . \end{aligned}$$

The last inequality holds for any $\lambda >0$, so we obtain

$$\begin{aligned} \mathbf {I}_1 + \mathbf {I}_2 \leqq C(N,\alpha ) \Vert v\Vert ^{1-\alpha }_{L^p(\mathbb {R}^N)} \Vert \nabla v\Vert ^\alpha _{ L^p(\mathbb {R}^N)} . \end{aligned}$$

(5.6)

By (5.3) and (5.6), we complete the proof of Lemma 11. $\quad \square$

Lemma 12

Let $\theta \in (0,1)$, and $N\geqq 1$. Let $\alpha _1, \alpha _2 \in (0,1)$ be such that $\alpha _1 < \alpha _2\theta$. Assume that $\Gamma$ is a $\theta$-Hölder continuous function on $\mathbb {R }$. Then we have

$$\begin{aligned} \Vert \nabla ^{\alpha _1} \Gamma (v) \Vert _{L^r(\mathbb {R}^N)} \leqq C \Vert v\Vert _{\dot{H} ^{\alpha _2}(\mathbb {R}^N)} , \end{aligned}$$

where

$$\begin{aligned} \frac{\alpha _1}{\theta } + \frac{N}{2} = \alpha _2 + \frac{N}{r} . \end{aligned}$$

(5.7)

Remark 11

Note that it follows from (5.7) that $r>2$.

Proof

The proof of Lemma 12 can be found in [3, Lemma 6.6] . $\quad \square$