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Local existence and blow-up criterion of the ideal density-dependent flows
Boundary Value Problems volume 2016, Article number: 101 (2016)
Abstract
In this paper, we consider two ideal density-dependent flows in a bounded domain, the Euler and magnetohydrodynamics equations. We prove the local existence and a blow-up criterion for each system.
1 Introduction
First, we consider the following 3D density-dependent Euler system:
Here Ω is a bounded domain with smooth boundary \(\partial\Omega \in C^{\infty}\), n is the outward unit normal to ∂Ω; the unknowns are the fluid velocity field \(\mathbf{u}=\mathbf {u}(x,t)\), the pressure \(\pi=\pi(x,t)\), and the density \(\rho=\rho(x,t)\).
Beirão da Veiga and Valli [1, 2] and Valli and Zajaczkowski [3] proved the unique solvability, local in time, in some supercritical Sobolev spaces and Hölder spaces in bounded domains. It is worth pointing out that in 1995 Berselli [4] discussed the standard ideal flow.
When \(\Omega:={\mathbb {R}^{3}}\), Danchin [5] and Danchin and Fanelli [6] (see also [7, 8]) proved the unique solvability, local in time, in some critical Besov spaces.
The first aim of this paper is to prove the local existence and a blow-up criterion of problem (1.1)-(1.5) in the \(L^{p}\) frame work. We will prove the following:
Theorem 1.1
Let \(0<\inf\rho_{0}\leq\sup\rho_{0}<\infty\), \(\rho_{0},\mathbf{u}_{0}\in W^{s,p}(\Omega)\) with integer \(s\geq3\), \(s>1+\frac{3}{p}\), and \(2< p<\infty \), and \(\operatorname {div}\mathbf{u}_{0}=0\) and \(\mathbf{u}_{0}\cdot\mathbf{n}=0\) on ∂Ω. Then there exists a positive time \(T^{*}>0\) such that problem (1.1)-(1.5) has a unique solution \((\rho,\mathbf {u})\) satisfying
Furthermore, if u satisfies
with \(0< T<\infty\), then the solution \((\rho,\mathbf{u},\pi)\) can be extended beyond \(T>0\).
Remark 1.1
When \(1< p\leq2\), we can prove a similar result.
We also consider the following ideal density-dependent MHD system:
Here Ω is a bounded domain with smooth boundary \(\partial\Omega \in C^{\infty}\), n is the outward unit normal to ∂Ω, and the unknowns are the plasma velocity \(\mathbf{u}=\mathbf {u}(x,t)\), the magnetic field \(\mathbf{b}=\mathbf{b}(x,t)\), the pressure \(\pi=\pi(x,t)\), and the density \(\rho=\rho(x,t)\). When \(\mathbf {b}=0\), system (1.8)-(1.13) reduces to the density-dependent Euler equations (1.1)-(1.5). When \(\Omega :={\mathbb {R}^{3}}\), Zhou and Fan [9] proved the local well-posedness of problem (1.8)-(1.13). For other related works, we refer to [10–14] and references therein.
In 1993, Secchi [15] was the first one to consider problem (1.8)-(1.13) and proved the local unique solvability with the main condition that
The second aim of this paper is to prove the local well-posedness of problem (1.8)-(1.13) without any smallness condition; furthermore, we will also prove a regularity criterion. We will prove the following:
Theorem 1.2
Let \(0<\inf\rho_{0}\leq\sup\rho_{0}<\infty\), \(\rho_{0},\mathbf{u}_{0},\mathbf {b}_{0}\in H^{s}\) with integer \(s\geq3\), \(\operatorname {div}\mathbf{u}_{0}=\operatorname {div}\mathbf {b}_{0}=0\) in Ω, and \(\mathbf{u}_{0}\cdot\mathbf{n}=\mathbf {b}_{0}\cdot\mathbf{n}=0\) on ∂Ω.
Then there exists a positive time \(T^{*}>0\) such that problem (1.8)-(1.13) has a unique solution \((\rho,\mathbf{u},\mathbf {b})\) satisfying
Furthermore, if u and b satisfy
with \(0< T<\infty\), then the solution \((\rho,\mathbf{u},\mathbf{b},\pi)\) can be extended beyond \(T>0\).
Remark 1.2
We are unable to prove Theorem 1.1 for the ideal density-dependent MHD system.
We will use the following well-known Osgood lemma in [16].
Lemma 1.3
(Osgood lemma)
Let y be a measurable positive function, f a positive, locally integrable function, and g a continuous increasing function. Assume that, for a positive real number a, the function y satisfies
If a is different from zero, then we have
If a is zero and \(g(s)\) satisfies \(\int_{0}^{1}\frac{dr}{g(r)}=+\infty\), then the function y is identically zero.
We will also use the following bilinear commutator and the product estimate:
-
(i)
If \(f\in W^{s,p}(\Omega)\cap C^{1}(\Omega)\) and \(g\in W^{s-1,p}(\Omega)\cap C(\Omega)\), then, for \(|\alpha|\leq s\),
$$\begin{aligned} \bigl\| D^{\alpha}(fg)-fD^{\alpha}g\bigr\| _{L^{p}(\Omega)}\leq C\bigl(\|f \|_{W^{s,p_{1}}(\Omega )}\|g\|_{L^{q_{1}}(\Omega)}+\|\nabla f\|_{L^{p_{2}}(\Omega)}\|g\| _{W^{s-1,q_{2}}(\Omega)}\bigr). \end{aligned}$$(1.17) -
(ii)
If \(f,g\in W^{s,p}(\Omega)\cap C(\Omega)\), then, for \(|\alpha|\leq s\),
$$ \bigl\| D^{\alpha}(fg)\bigr\| _{L^{p}(\Omega)}\leq C\bigl(\|f\|_{W^{s,p_{1}}(\Omega)}\|g\| _{L^{q_{1}}(\Omega)}+\|f\|_{L^{p_{2}}(\Omega)}\|g\|_{W^{s,q_{2}}(\Omega )}\bigr) $$(1.18)with integer \(s>0\), \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac {1}{p_{2}}+\frac{1}{q_{2}}\), and \(1< p<\infty\).
The case with \(p=2\), \(p_{1}=q_{2}=p\), \(q_{1}=p_{2}=\infty\) has been proved in [17]. Since the proof of (1.18) is similar to that of (1.17), we will prove (1.17) only in the Appendix.
2 Local existence of the Euler system
This section is devoted to the proof of local existence for the Euler system. We only need to prove a priori estimates (1.6).
First, by the maximum principle, we have the well-known estimates
Testing (1.2) by u and using (1.1), (1.3), and (1.4), we see that
Applying \(D^{s}\) to (1.1), testing by \(|D^{s}\rho|^{p-2}D^{s}\rho\), and using (1.3), (1.4), and (1.17), we derive
Using (1.1), we rewrite (1.2) as follows:
Applying \(D^{s}\) to (2.4), testing by \(|D^{s}u|^{p-2}D^{s}u\), and using (1.3), (1.4), (1.17), (1.18), and (2.1), we deduce that
Testing (2.4) by ∇π and using (1.3), (1.4), (2.1), and (2.2), we infer that
Taking div to (2.4), we observe that
Using (1.1), (1.2), and (1.4), we deduce that
Using (1.18) and the well-known \(W^{s,p}\)-estimates of problem (2.7)-(2.8) [18], we have
where we used the estimate [18]
By the Gagliardo-Nirenberg inequality
it follows from (2.6), (2.9), and (2.10) that
Combining (2.3), (2.5), and (2.11) and using Osgood’s lemma (for some T) and the inequalities
we arrive at
This completes the proof.
3 A blow-up criterion for the Euler system
This section is devoted to the proof of regularity criterion for the Euler system. We only need to establish a priori estimates.
First, we still have (2.1) and (2.2).
Taking ∇ to (1.1), testing by \(|\nabla\rho|^{p-2}\nabla\rho \). and using (1.3) and (1.4), we derive
whence
Integrating this inequality and taking the limit as \(p\rightarrow+\infty \), we have
It follows from (2.6) that
It follows from (2.7), (2.8), (1.7), (3.1), (3.2), and the \(W^{2,p}\)-estimates of problem (2.7)-(2.8) that
for any \(3< p<\infty\), and thus
Similarly to (2.9), we have
and thus
Combining (2.3), (2.5), (3.4), (1.7), (3.3), and (3.1) and using the Gronwall inequality, we arrive at (2.12).
This completes the proof.
4 Local existence for the MHD system
This section is devoted to the proof of local existence for the MHD system. We only need to prove a priori estimates (1.15). Before going to detailed estimates, we write the case with \(p=2\), \(p_{1}=q_{2}=p\), \(q_{1}=p_{2}=\infty\) in (1.17) and (1.18) as follows:
-
(i)
If \(f,g\in H^{s}(\Omega)\cap C(\Omega)\), then
$$ \|fg\|_{H^{s}(\Omega)}\leq C\bigl(\|f\|_{H^{s}(\Omega)}\|g\|_{L^{\infty}(\Omega )}+\|f \|_{L^{\infty}(\Omega)}\|g\|_{H^{s}(\Omega)}\bigr). $$(4.1) -
(ii)
If \(f\in H^{s}(\Omega)\cap C^{1}(\Omega)\) and \(g\in H^{s-1}(\Omega)\cap C(\Omega)\), then, for \(|\alpha|\leq s\),
$$ \bigl\| D^{\alpha}(fg)-fD^{\alpha}g\bigr\| _{L^{2}(\Omega)}\leq C\bigl(\|f \|_{H^{s}(\Omega)}\| g\|_{L^{\infty}(\Omega)}+\|f\|_{W^{1,\infty}(\Omega)}\|g \|_{H^{s-1}(\Omega )}\bigr). $$(4.2)
First, by the maximum principle we have the well-known estimates
Testing (1.2) by u and using (1.8) and (1.11), we see that
Testing (1.10) by b and using (1.4), we find that
Summing up (4.4) and (4.5) and noting the cancellation of the terms on the right-hand sides of (4.4) and (4.5), we get
Applying \(D^{s}\) to (1.8), testing by \(D^{s}\rho\), and using (1.11) and (4.2), we derive
Applying \(D^{s}\) to (1.9), testing by \(D^{s}\mathbf{u}\), and using (1.11), we get
Applying \(D^{s}\) to (1.10), testing by \(D^{s}\mathbf{b}\), and using (1.11), we deduce
Summing up (4.8) and (4.9) and noting that \(I_{2}+I_{7}=0\), we find that
Using (4.2) and (4.1), we bound \(I_{1}\), \(I_{3}\), \(I_{4}\), \(I_{5}\), \(I_{6}\), and \(I_{8}\) as follows:
Inserting these estimates into (4.10), we have
Testing (1.9) by \(\partial_{t}\mathbf{u}\) and using (1.11), we find that
whence
Applying \(D^{s-1}\) to (1.9), testing by \(D^{s-1}\partial_{t}\mathbf {u}\), and using (1.8), we have
whence
Using (4.1) and (4.2) again, we bound \(J_{1}\), \(J_{2}\), and \(J_{3}\) as follows:
for any \(0<\epsilon<1\).
Inserting these estimates into (4.12) and (4.13) and taking ϵ small enough, we have
Using (1.8) and (1.11) and setting \(\tilde{\pi}:=\pi+\frac{1}{2} \mathbf{b}^{2}\), we rewrite (1.9) as
Testing (4.15) by ∇π̃ and using (1.11) and (4.3), we infer that
Using (1.8), (1.9), and (1.12), we deduce that
Taking div to (4.15), we observe that
Using (4.1) and the well-known \(H^{s+1}\)-estimates of problems (4.18) and (4.17) [18], we have
whence
where we used the Gagliardo-Nirenberg inequality
and the well-known estimate [18]
Combining (4.7), (4.11), (4.14), and (4.20) and using the Osgood lemma, we arrive at (1.15).
This completes the proof.
5 A blow-up criterion for the MHD system
This section is devoted to the proof of regularity criterion for the MHD system. We only need to establish a priori estimates.
First, we still have (4.3) and (4.6).
Taking ∇ to (1.8), testing by \(|\nabla\rho|^{p-2}\nabla \rho\), and using (1.11) and (1.16), we derive
whence
Integrating this inequality and taking the limits as \(p\rightarrow +\infty\), we have
It follows from (4.6) and (1.16) that
Similarly to (4.16), we find that
It follows from (4.17), (4.18), (5.1), (5.2), (5.3), and the \(W^{2,p}\)-estimates of problem (4.17)-(4.18) that
for any \(3< p<\infty\), and thus
It follows from (4.15), (4.3), and (5.4) that
Similarly to (4.19), we have
whence
We still have (4.13), and similarly to (4.14), we have
which gives
Similarly to (4.7), we have
We still have (4.10). We bound \(I_{1}\), \(I_{3}\), \(I_{4}\), \(I_{5}\), \(I_{6}\), and \(I_{8}\) as follows:
Inserting these estimates into (4.10) and using (5.5) and the Gronwall inequality, we conclude that
This completes the proof.
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Acknowledgements
This paper is partially supported by NSFC (Nos. 11171154 and 71102145). The authors are indebted to the referees for careful reading and helpful comments.
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Appendix:  Proof of (1.17)
Appendix:  Proof of (1.17)
We only prove the case \(|\alpha|=s\). We have
We will use the following two Gagliardo-Nirenberg inequalities:
with \(i-\frac{d}{p_{i}}=(1-\alpha_{i}) (1-\frac{d}{p_{2}} )+\alpha _{i} (s-\frac{d}{p_{1}} )\), where d is the dimension number.
Inserting (A.2) and (A.3) into (A.1) and using the Young inequality give (1.17).
This completes the proof.
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He, F., Fan, J. & Zhou, Y. Local existence and blow-up criterion of the ideal density-dependent flows. Bound Value Probl 2016, 101 (2016). https://doi.org/10.1186/s13661-016-0610-y
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DOI: https://doi.org/10.1186/s13661-016-0610-y
MSC
- 35Q35
- 76D03
Keywords
- Euler
- ideal MHD
- local existence
- blow-up criterion