Abstract
Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.
Similar content being viewed by others
1 Introduction and the main results
The evolutionary equation related to the p-Laplacian
arises in the fields of mechanics, physics and biology. For instance, in the theory of non-Newtonian fluids, the quantity p is a characteristic of the medium, the media with \(p > 2\) are called dilatant fluids and those with \(p < 2\) are called pseudoplastics; if \(p=2\) they are Newtonian fluids. If \(a(x)\equiv1\), there is a tremendous amount of work on the existence, the uniqueness and the regularity of the weak solutions of the equation, one can refer to Refs. [1,2,3,4,5,6,7] and the references therein. Zhao [8] had studied the equation
and revealed some essential differences coming from the term \(f(\nabla u, u,x,t)\). Yin–Wang [9] had studied the equation
revealed how the degeneracy of the diffusion coefficient \(a(x)\) affects the boundary value condition, where \(D_{i} =\frac{\partial}{\partial x_{i}}\), \(a \in C(\overline{\varOmega})\) and \(a(x)\geq0\).
In this paper, we consider
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with appropriately smooth boundary, \(p>1\), \({Q_{T}} = \varOmega \times(0,T)\), \(a(x) \in C^{1}(\overline{\varOmega})\), \(a(x)\geq0\) and
the nonlinear convection \(b_{i}(s,x,t)\in C(\mathbb{R}\times\overline {Q_{T}})\), the source term \(f(s,x,t)\in C(\mathbb{R}\times\overline {Q_{T}})\). Comparing with [9], we must pay attention on how these two nonlinear terms affect the well-posedness problem of Eq. (1.4).
The condition (1.5) guarantees that Eq. (1.4) has not hyperbolic character. In other words, if the set \(\{x\in\varOmega: a(x)=0\}\) has an interior point, then Eq. (1.4) is with a hyperbolic-parabolic type, the uniqueness of the solution may be obtained only in the sense of the entropy solution. Since the condition (1.5), \(a(x)\) only may be degenerate on the boundary, Eq. (1.4) is a sheer degenerate parabolic equation. Thus, we can discuss its well-posedness of the usual weak solutions instead of the entropy solution.
Drawing on the experience of the linear degenerate parabolic theory, to study the well-posedness of the solutions of Eq. (1.4), the initial value
is always necessary. While, the usual Dirichlet boundary value condition
may be overdetermined. So it is only a partial boundary condition
imposed in [9], where \(\varSigma_{p}\subseteq\partial\varOmega\). In particular, if \(\varSigma_{p}=\varnothing\), then there is not any boundary value condition. But the partial boundary condition [9] is in a weaker sense than the trace.
The methods used in what follows are different from those in [9], we still use the sense of the trace to define the boundary value condition (1.7) or (1.8). Roughly speaking, we will show that the condition
can substitute the boundary value condition (1.7). But if (1.9) is not right, only the partial boundary value condition (1.8) is required, we need to find the explicit formulas of \(\varSigma_{p}\) and judge which one is the best.
The definition of the weak solutions follows a Banach space which is defined as follows. For every fixed \(t\in[0, T]\), let
and denote by \(V'_{t}(\varOmega)\) its dual space. By \(\mathbf{W}(Q_{T})\) we denote the Banach space
\(\mathbf{W}'(Q_{T})\) is the dual space of \(\mathbf{W}(Q_{T})\) (the space of linear functionals over \(\mathbf{W}(Q_{T})\)).
Definition 1.1
A function \(u(x,t)\) is said to be a weak solution of Eq. (1.4) with the initial value (1.6), if
and for any function \(\varphi_{1} \in L^{\infty}(0,T; W_{0}^{1,p}(\varOmega ))\), \(\varphi_{2}\in L^{1}(0,T; W^{1,p}_{\mathrm{loc}}(\varOmega))\),
The initial value is satisfied in the sense of that
Definition 1.2
Let \(p>1\). The function \(u(x,t)\) is said to be the weak solution of Eq. (1.4) with the initial boundary values (1.6)–(1.7) (or (1.8)) if u satisfies Definition 1.1, and the boundary condition (1.7) (or (1.8) respectively) is satisfied in the sense of trace.
Theorem 1.3
If \(p>1\), \(b_{i}(s,x,t)\) and \(f(s,x,t)\) are \(C^{1}(\mathbb{R}\times\overline{\varOmega}\times[0,T])\) functions, and
then Eq. (1.4) with initial value (1.6) has a weak solution.
Theorem 1.4
Let \(p>1\), \(\int_{\varOmega}a(x)^{-\frac {1}{p-1}}\,dx<\infty\), \(b_{i}(s, x, t)\) and \(f(s,x,t)\) be \(C(\mathbb {R}\times\overline{\varOmega}\times[0,T])\) functions. Then the initial boundary value problem (1.4)–(1.6) and (1.7) (or (1.8)) has a solution.
The first aim of this paper is to prove the following stability theorems without any boundary value condition.
Theorem 1.5
Let \(u(x,t)\), \(v(x,t)\) be two solutions of (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively. If there is a function \(g_{i}(x)\) with \(g_{i}(x)|_{x\in\partial\varOmega}=0\) such that
\(f(s,x,t)\) is a Lipschitz function and \(a(x)\) satisfies
then
The stability (1.17) is true.
Here and the hereafter, for any positive small \(\delta>0\), \(\varOmega _{\delta}=\{x\in\varOmega: a(x)>\delta\}\).
An interesting corollary from Theorem 1.5 is that, if \(\int_{\varOmega }a(x)^{-\frac{1}{p-1}}\,dx<\infty\), then without the condition (1.16), only if the condition (1.17) holds, the stability (1.18) is true. Additionally, the second inequality of (1.16) implies that \(g_{i}(x)|_{x\in\partial\varOmega}=0\). In fact, the condition (1.17) can be replaced by the other conditions. The following theorem is one of results expected.
Theorem 1.6
Let \(u(x,t)\), \(v(x,t)\) be two solutions of (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively. If there is a function \(g_{i}(x)\) with \(g_{i}(x)|_{x\in\partial\varOmega}=0\) such that (1.16) is true \(f(s,x,t)\) is a Lipschitz function and \(a(x)\) satisfies
then the stability (1.18) is true.
Moreover, by choosing suitable test function, using the Hölder inequality, we can prove another stability theorem without any boundary value condition.
Theorem 1.7
Let \(u(x,t)\), \(v(x,t)\) be two solutions of (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively. If \(f(s,x,t)\) and \(b_{i}(s,x,t)\) are Lipschitz functions, \(a(x)\) satisfies
then the stability (1.18) is true.
The second aim of this paper is to prove the stability theorems based on the partial boundary value condition (1.8).
Theorem 1.8
Let \(u(x,t)\), \(v(x,t)\) be two solutions of (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively, and with the same partial boundary value condition
where
If \(a(x)\) satisfies (1.5) and
\(f(s,x,t)\) and \(b_{i}(s,x,t)\) are Lipschitz functions, then the stability (1.18) is true.
We emphasize that the conditions (1.16), (1.17), (1.19), (1.20) and (1.23) are used to prove the stability of the weak solutions. In fact, only if \(a(x)\) satisfies (1.5), the uniqueness is always true.
Theorem 1.9
Let \(p>1\), \(b_{i}(s, x, t)\) be a Lipschitz function, \(f(s,x,t)\) is a continuous function. If \(u(x,t)\), \(v(x,t)\) are two solutions of Eq. (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively, with the same partial boundary value condition (1.21) in which
then there exists a positive constant \(\beta\geq2\) such that
In particular, for any small enough constant \(\delta>0\),
where \(\varOmega_{\delta}=\{x\in\varOmega: a(x)>\delta\}\) as before.
From this theorem, if \(u_{0}(x)=v_{0}(x)\), by the arbitrariness of δ in (1.26), the solution of Eq. (1.4) with the initial value and the partial boundary value condition (1.24) is unique. We can see that, if \(\varSigma_{2}=\partial\varOmega\), i.e., \(a(x)\geq c>0\), Eq. (1.4) is similar to the classical evolutionary p-Laplacian equation, (1.24) is the usual Dirichlet boundary value condition, the uniqueness is true naturally. While \(\varSigma_{2}=\emptyset\), i.e., \(a(x)=0\) on the boundary ∂Ω, the uniqueness of the weak solution is true independent of the boundary value condition. If \(b_{i}(u,x,t)=b_{i}(u)\), \(u_{t}\in L^{2}(Q_{T})\) and \(a(x)=d^{\alpha}(x)\) where \(d(x)=\operatorname {dist}(x, \partial\varOmega)\), the same conclusion as of Theorem 1.9 had been proved by the author in his previous work [10]. So, essential progress of this paper is that we do not assume that \(a(x)|_{x\in \partial\varOmega}=0\), the best partial boundary value condition is (1.24). This fact also remains an open problem: whether the partial boundary value condition (1.21) can be replaced by (1.24).
2 The existence of the weak solutions
This section considers the weak solution of the initial-value problem for Eq. (1.4). It is supposed that \(u_{0}\) satisfies (1.15)
By the results of [10, Sect. 8], also referring to [11], we have the following important lemma.
Lemma 2.1
If \(u_{\varepsilon}\in L^{\infty}(0,T;L^{2}(\varOmega ))\cap\mathbf{W}(Q_{T})\), \(\Vert u_{\varepsilon t} \Vert _{\mathbf{W}'(Q_{T})}\leq c\), \(\|\nabla(|u_{\varepsilon }|^{q-1}u_{\varepsilon})\|_{p,Q_{T}}\leq c\), then there is a subsequence of \(\{u_{\varepsilon}\}\) which is relatively compactness in \(L^{s}(Q_{T})\) with \(s\in(1,\infty)\). Here \(q\geq1\).
We consider the following regularized problem:
For any \({u_{\varepsilon,0}} \in{C^{\infty}_{0} }(\varOmega)\), \({a(x) \vert {\nabla{u_{\varepsilon,0}}} \vert ^{p}}\) uniformly is convergent to \(a(x)|\nabla u_{0}(x)|^{p}\) in \({L^{1}}(\varOmega)\), it is well known that the above problem has a unique classical solution [12, 13].
According to the maximum principle [2], there is a constant c only dependent on \(\Vert {{u_{0}}} \Vert _{{L^{\infty}}(\varOmega)}\) but independent on ε, such that
Multiplying (2.1) by \(u_{\varepsilon}\) and integrating it over \(Q_{T}\), we easily have
For small enough \(\delta>0\), since \(p>1\), by (1.5) and (2.5),
Also
Now, for any \(v\in\mathbf{W}(Q_{T})\), \(\|v\|_{W(Q_{T})}=1\),
Using the Young inequality, we can show that
then
Now, let \(\varphi\in C_{0}^{1}(\varOmega)\), \(0\leq\varphi\leq1\) such that
Then
we have
and so
By Lemma 2.1, \(\varphi u_{\varepsilon}\) is relatively compact in \(L^{s}(Q_{T})\) with \(s\in(1,\infty)\). Then \(\varphi u_{\varepsilon}\rightarrow\varphi u\) a.e. in \(Q_{T}\). In particular, due to the arbitrariness of δ, \(u_{\varepsilon}\rightarrow u\) a.e. in \(Q_{T}\).
Hence, by (2.4), (2.7), there exists a function u and an n-dimensional vector function \(\overrightarrow{\zeta}= ({\zeta_{1}}, \ldots,{\zeta_{n}})\) satisfying
and
Similar to the evolutionary p-Laplacian equation, we can prove that
for any function \(\varphi \in C_{0}^{1} ({Q_{T}})\). Then
If we denote \(\varOmega_{\varphi}=\operatorname{supp}\varphi\), then
Now, for any \(\varphi_{1}\in C_{0}^{1} ({Q_{T}})\), \(\varphi_{2}(x,t)\in L^{1}(0,T; W^{1,p}_{\mathrm{loc}}(\varOmega))\), it is clearly that
By the fact that \(C_{0}^{\infty}(\varOmega_{\varphi_{1}})\) is dense in \(W^{1, p}(\varOmega_{\varphi_{1}})\), by a limit process, we have
which implies that
Again by a limit process, \(\varphi_{1}\) can be chosen as in Definition 1.1.
At last, we are able to prove (1.14) as in [14], then u is a solution of Eq. (1.4) with the initial value (1.6) in the sense of Definition 1.1. Thus we have Theorem 1.3. By a similar method to [15], one easily proves the following lemma, we omit the details here.
Lemma 2.2
If \(\int_{\varOmega}a(x)^{-\frac{1}{p-1}}\,dx<\infty \), u is a weak solution of Eq. (1.4) with the initial value (1.6). Then, for any given \(t\in[0,T)\), \(u\in W^{1,\gamma}(\varOmega)\) for some \(\gamma>1\), and the trace of u on the boundary ∂Ω can be defined in the traditional way.
3 The stability without the boundary value condition
For any given positive integer n, let \({g_{n}}(s)\) be an odd function, and
Clearly, if denoting \(G_{n}(s)=\int_{0}^{s}g_{n}(s)\,ds\), then
and by
we have
where c is independent of n.
Lemma 3.1
Let \(u\in\mathbf{W}(Q_{T})\), \(u_{t}\in\mathbf {W}'(Q_{T})\). Then ∀ a.e. \(t_{1}, t_{2}\in(0, T)\),
This is Corollary 2.1 of [11].
By a similar analysis, one can generalize Lemma 3.1.
Lemma 3.2
Let \(u\in\mathbf{W}(Q_{T})\), \(u_{t}\in\mathbf {W}'(Q_{T})\). For any continuous function \(h(s)\), \(H(s)=\int_{0}^{s}h(s)\,ds\), a.e. \(t_{1}, t_{2}\in(0, T)\),
Proof of Theorem 1.5
Let \(u(x,t)\) and \(v(x,t)\) be two weak solutions of Eq. (1.4) with the initial values \(u(x,0)\), \(v(x,0)\), respectively. Let
By a limit process, we can choose \(\chi_{[\tau,s]}\phi_{n}{g_{n}}(u - v)\) as the test function, where \(\chi_{[\tau,s]}\) is the characteristic function of \([\tau, s]\subset(0, T)\), then
In the first place,
By Lemma 3.2, using the Lebesgue dominated convergence theorem,
Since \(\nabla\phi_{n}=n\nabla a(x)\) when \(x\in\varOmega\setminus\varOmega _{\frac{1}{n}}\), in the other places, it is identical to zero, by the assumption of (1.19), we have
Since \(a(x)\in C^{1}(\overline{\varOmega})\), by (3.5),
In the second place, since \(b_{i}(s,x,t)\) satisfies the condition (1.16)
using the Lebesgue dominated convergence theorem, we have
Last but not least, by condition (1.18) and \(g_{i}(x)|_{x\in\partial \varOmega}=0\), we clearly have
Now, let \(n\rightarrow\infty\) in (3.2). Then
where \(l\leq1\)
By (3.10), we easily to get
and by the arbitrariness of τ, we have
□
4 Proofs of Theorem 1.6 and Theorem 1.7
Proof of Theorem 1.6
Let \(u(x,t)\) and \(v(x,t)\) be two weak solutions of Eq. (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively.
For large enough m, let
By a limit process, we can choose \(\chi_{[\tau,s]}{g_{n}}(\phi_{m}(u - v))\) as the test function, then
Certainly, we still have
By Lemma 3.2, using the Lebesgue dominated convergence theorem,
As before, \(\nabla\phi_{m}=m\nabla a(x)\) when \(x\in\varOmega\setminus \varOmega_{\frac{1}{m}}\), in the other places, it is identical to zero, by the fact that
using the Lebesgue dominated convergence theorem, we have
In the second place, since \(b_{i}(s,x,t)\) satisfies the condition (1.18)
using the Lebesgue dominated convergence theorem, we have
Last but not least, by
using the Lebesgue dominated convergence theorem, we have
Now, let \(n\rightarrow\infty\) in (5.2). Then
where \(l\leq1\).
By (4.8), we easily get
and by the arbitrariness of τ, we have
□
Proof of Theorem 1.7
Let \(u(x,t)\) and \(v(x,t)\) be two weak solutions of Eq. (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively. Let
By a limit process, we can choose \(\chi_{[\tau,s]}{g_{n}}(\varphi _{m}(u - v))\) as the test function, then
Similarly, we have
and
As before, \(\nabla\phi_{m}=m\nabla a(x)\) when \(x\in\varOmega\setminus \varOmega_{\frac{1}{m}}\), in the other places, it is identical to zero. Since \(\int_{\varOmega}a(x)|\nabla u|^{p}\,dx<\infty\), by that \(a(x)>0\) when \(x\in\varOmega\), we have
which yields
By the fact that
and the assumption of (1.20), using the Lebesgue dominated convergence theorem, we have
Since \(b_{i}(s,x,t)\) is a Lipschitz function, using Lebesgue dominated convergence theorem, we have
obviously.
Last but not least, by (1.21) using the Lebesgue dominated convergence theorem, we have
Now, after letting \(n\rightarrow\infty\), let \(m\rightarrow\infty\) in (4.10). Then we have the conclusion. □
5 The usual boundary value condition
By Lemma 2.2, if \(\int_{\varOmega}a(x)^{-\frac{1}{p-1}}\,dx<\infty\), then we can define the trace of u on the boundary ∂Ω. If one imposes the usual boundary value condition (1.7), the stability of the weak solutions is true. For the completeness of the paper, we also give this conclusion and its proof here.
Theorem 5.1
Let \(p>1\), \(\int_{\varOmega}a(x)^{-\frac {1}{p-1}}\,dx<\infty\), \(f(s,x,t)\) and \(b_{i}(s,x,t)\) be Lipschitz functions. If \(u(x,t)\), \(v(x,t)\) are two solutions of Eq. (1.4) with the usual homogeneous value condition,
and with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively, then
Proof
By a limit process, we can choose \(\chi_{[\tau ,s]}{g_{n}}(u - v)\) as a test function. Then
As usual, we have
By Lemma 3.2,
Moreover, similar to [15], we can prove that
Now, let \(n\rightarrow\infty\) in (5.2). Then
Let \(\tau\rightarrow0\). Then, by the Gronwall inequality, we have
Theorem 5.1 is proved. □
The interesting problem is that, since \(a(x)\) may be degenerate on the boundary, the usual boundary value condition (1.7) is overdetermined [9]. Obviously, Theorem 1.8 has solved this problem partially.
Proof of Theorem 1.8
If \(u(x,t)\), \(v(x,t)\) are two solutions of (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively, and with the same partial boundary value condition (1.21)
Let \(\phi_{n}(x)\) be defined as in the proof of Theorem 1.5. By the assumption
where
we can choose \(\chi_{[\tau,s]}\phi_{n}(x){g_{n}}(u - v)\) as a test function. By the condition (1.23), similar to the proof Theorem 1.5, we are able to show the conclusion of Theorem 1.8, we omit the details here. □
6 The uniqueness of the solution
In this section, we will prove Theorem 1.9.
Proof of Theorem 1.9
Let \(u(x,t)\), \(v(x,t)\) be two solutions of Eq. (1.4) with the initial values \(u_{0}(x)\), \(v_{0}(x)\), respectively, and with
where
Then we may choose \(\chi_{[\tau,s]}(u-v)a^{\beta}\) as a test function, where \(\beta\geq1\) is a constant,
where \(Q_{\tau,s}=\varOmega\times(\tau,s)\). By that \(|\nabla a|< c\), and
by the Hölder inequality, we can show that
where \(l\geq1\). Here, we have the fact that
due to \(\beta\geq1\).
As for the convection term,
Since \(\beta\geq2\), \(|a_{x_{i}}|\leq|\nabla a|\leq c\), by the Hölder inequality, we have
and
Since \(f(s,x,t)\) is a continuous function, \(\|u\|_{L^{\infty}(Q_{T})}\leq c\), \(\|u\|_{L^{\infty}(Q_{T})}\leq c\),
obviously.
By Lemma 3.1,
From (6.2)–(6.7), by (6.1), we have
where \(l\leq1\). By (6.8), we easily get
and by the arbitrariness of τ, we have
□
7 Conclusion
Compared with our previous work [10], not only is the equation considered in this paper more general, but also the conclusions are much better. In particular, the uniqueness of the weak solution based on a partial boundary value condition (Theorem 1.9) is always true. This conclusion is more or less beyond one’s imagination. Benedikt et al. had considered the equation
and showed that the uniqueness of the solution is not true [16]. But Theorem 1.9 in this paper implies that, only if \(u_{t}\in W'(Q_{T})\), the uniqueness of the solution to Eq. (7.1) is true.
References
DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)
Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations. World Scientific, Singapore (2001)
Lee, K., Petrosyan, A., Vazquez, J.L.: Large time geometric properties of solutions of the evolution p-Laplacian equation. J. Differ. Equ. 229, 389–411 (2006)
Kalashnikov, A.S.: Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations. Russ. Math. Surv. 42(2), 169–222 (1987)
Nakao, N.: \(L^{p}\) estimates of solutions of some nonlinear degenerate diffusion equation. J. Math. Soc. Jpn. 37, 41–63 (1985)
Nabana, E.: Uniqueness for positive solutions of p-Laplacian problem in an annulus. Ann. Fac. Sci. Toulouse Math. 8, 143–154 (1999)
Zhan, H.: Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Appl. Math. 53, 521–533 (2008)
Zhao, J.N.: Existence and nonexistence of solutions for \({u_{t}} =\operatorname{div}({ \vert {\nabla u} \vert ^{p - 2}}\nabla u) + f(\nabla u,u,x,t)\). J. Math. Anal. Appl. 172(1), 130–146 (1993)
Yin, J., Wang, C.: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differ. Equ. 237, 421–445 (2007)
Zhan, H.: The uniqueness of a nonlinear diffusion equation related to the p-Laplacian. J. Inequal. Appl. 2018, 7 (2018). https://doi.org/10.1186/s13660-017-1596-4
Antontsev, S.N., Shmarev, S.I.: Parabolic equations with double variable nonlinearities. Math. Comput. Simul. 81, 2018–2032 (2011)
Gu, L.: Second Order Parabolic Partial Differential Equations. The Publishing Company of Xiamen University, Xiamen (2002) (in Chinese)
Taylor, E.M.: Partial Differential Equations III. Springer, Berlin (1999)
Zhan, H.: The solution of convection–diffusion equation. Chin. Ann. Math. 34(2), 235–256 (2013) (in Chinese)
Zhan, H., Yuan, H.: Diffusion convection equation with boundary degeneracy. J. Jilin Univ. Sci. Ed. 53(3), 353–358 (2015) (in Chinese)
Benedikt, J., Bobkov, V.E., Girg, P., Kotrla, L., Takac, P.: Nonuniqueness of solutions of initial-value problems for parabolic p-Laplacian. Electron. J. Differ. Equ. 2015, 38 (2015)
Simon, J.: Compact sets in the space \(L^{p}(0,t;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1952)
Acknowledgements
The author would like to thank everyone for their help!
Availability of data and materials
No applicable.
Funding
The paper is supported by Natural Science Foundation of Fujian province, supported by Science Foundation of Xiamen University of Technology, China.
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares to have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhan, H. Hölder inequality applied on a non-Newtonian fluid equation with a nonlinear convection term and a source term. J Inequal Appl 2018, 344 (2018). https://doi.org/10.1186/s13660-018-1938-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1938-x