On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

Abstract

We prove an existence and uniqueness of weak solutions of a general mean field equation of nonlinear diffusion with the Newtonian potential pressure. Moreover, we study the finite speed of propagation of solutions.

Appendix

Lemma 4.1

Let $v_0\in L^1({\mathbb {R}}^N)\cap L^\infty ({\mathbb {R}}^N)$. Assume that for any $1<q<\infty$, v(x, t) satisfies

$$\begin{aligned} \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \le \Vert v_0\Vert _{L^q({\mathbb {R}}^N)},\quad \text {for } \, t\in (0,T). \end{aligned}$$

Then, we have

$$\begin{aligned} \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)} \le \Vert v_0\Vert _{L^\infty ({\mathbb {R}}^N)},\quad \text {for } \, t\in (0,T)\,. \end{aligned}$$

(4.1)

Proof of Lemma 4.1

We first claim that $v(.,t)\in L^\infty ({\mathbb {R}}^N)$ for $t\in (0,T)$. Indeed, assume a contradiction that there is a $t_0\in (0,T)$ such that

$$\begin{aligned} \Vert v(.,t_0)\Vert _{L^\infty ({\mathbb {R}}^N)}=+\infty \,. \end{aligned}$$

(4.2)

Now, let us set $A_n = \{x\in {\mathbb {R}}^N: |v(x,t_0)| > n\}$ for $n\ge 1$. For $1<q<\infty$, it is obvious that

$$\begin{aligned} n|A_n|^{1/q} \le \Vert v(.,t_0)\Vert _{L^q({\mathbb {R}}^N)}\le \Vert v_0\Vert _{L^q({\mathbb {R}}^N)} \le \Vert v_0\Vert _X . \end{aligned}$$

Hence, we get $\displaystyle \lim \nolimits _{n\rightarrow \infty } |A_n| = 0$. This contradicts to (4.2).

Thus, we obtain the claim.

Next, we show that for any $t\in (0,T)$,

$$\begin{aligned} \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)}\rightarrow \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)} ,\,\text { when } \, q\rightarrow \infty \,. \end{aligned}$$

(4.3)

Fix $q_0>1$. Thanks to the interpolation inequality, for any $q\in (q_0,\infty )$ we have

$$\begin{aligned} \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \le \Vert v(.,t)\Vert ^{1-\frac{q_0}{q}}_{L^\infty ({\mathbb {R}}^N)} \Vert v(.,t)\Vert ^{\frac{q_0}{q}}_{L^{q_0}({\mathbb {R}}^N)} , \end{aligned}$$

which yields

$$\begin{aligned} \limsup _{q\rightarrow \infty } \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \le \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)} \,. \end{aligned}$$

(4.4)

On the other hand, let $E_\lambda = \{ x\in {\mathbb {R}}^N: |v(x,t)|>\lambda \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)}\}$, for $\lambda \in (0,1)$. Then, we observe that $|E_\lambda |>0$, $\Vert v(.,t)\Vert _{L^{q_0}(E_\lambda )}>0$, and

$$\begin{aligned} \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \ge \big (\lambda \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)} \big )^{1-\frac{q_0}{q}} \Vert v(.,t)\Vert ^\frac{q_0}{q}_{L^{q_0}(E_\lambda )} , \end{aligned}$$

hence $\liminf \nolimits _{q\rightarrow \infty } \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \ge \lambda \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)}$.

Since the last inequality holds for $\lambda \in (0,1)$, then we obtain

$$\begin{aligned} \liminf _{q\rightarrow \infty } \Vert v(.,t)\Vert _{L^q({\mathbb {R}}^N)} \ge \Vert v(.,t)\Vert _{L^\infty ({\mathbb {R}}^N)} . \end{aligned}$$

Thus, (4.3) follows from (4.4) and the last inequality.

Finally, (4.1) is done by applying (4.3) to v(., t) and $v_0$. $\square$