ABSTRACT
In this work, we consider a initial-value problem for an doubly nonlinear advection-diffusion equation, and we present a critical value of κ up to wich the initial-value problem has global solution independent of the initial data u 0, and from which global solutions may still exists, but from initial data u 0 satisfying certain conditions. For this, we suppose that the function \(<mml:math><mml:mi mathvariant="bold-italic">f</mml:mi> <mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo> <mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi> <mml:mo>)</mml:mo></mml:math>\) in the advection term, writted in the divergent form, satisfies certain conditions about your variation in ℝn , and we also use the decrease of the norm and an control for the norm of solution .
Keywords:
doubly nonlinear parabolic equation; global solutions; conditions for global solutions
RESUMO
Neste trabalho, consideramos um problema de valor inicial para uma equação de advecção-difusão duplamente não linear, e apresentamos um valor crítico de κ até o qual o problema de valor inicial tem solução global independente do dado inicial u 0, e a partir do qual as soluções globais ainda podem existir, mas para dados iniciais u 0 satisfazendo a determinadas condições. Para isso, supomos que a função no termo advectivo, escrito na forma divergente, satisfaz a certas condições a respeito de sua variação em ℝn , e usamos também o decrescimento na norma e um controle para a norma da solução .
Palavras-chave:
equações parabólicas duplamente não lineares; soluções globais; condições para soluções globais
1 INTRODUCTION
We will considerer the regularized problem
where is fixed and is given; α and β are constants, with and ; and the function satisfies
for and for all , where denotes the variation of in ℝn , and controls the magnitude of its derivatives.
As solution to the regularized problem (1.1) in a determined interval , we consider a function , smooth, which solves the equation in (1.1) in the classic sense for , and satisfies the initial condition in the sense of . The interval is known as the maximal interval for the solution, and the existence of such T ∗ is guaranteed by general theory of Parabolic Equations (see, e.g., 77. O. Ladyzhenskaya, V.A. Solonnikov & N.N. Uralceva. “Linear and Quasilinear Equations of Parabolic Type”. American Mathematical Society, Providence (1968). or 88. D. Serre. “Systems of Conservation Laws”, volume 1. Cambridge University Press, Cambridge (1999).), so that the local existence of solutions for the problem (1.1) is guaranteed.
If we consider the additional stability condition
then we can obtain for the solutions of (1.1) the following fundamental supnorm estimate:
where the values of ρ and σ are and . Thereby, it is easy to see that the solution has global existence, and more: the solution goes to the zero when (see 22. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: um caso de decrescimento. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 5, v.1. SBMAC (2017). doi:10.5540/03.2017.005.01.0034.
https://doi.org/10.5540/03.2017.005.01.0...
).
However, in the general case, the existence of global solutions is not easy to obtain. The central question is that in the search for conditions that guarantee the global existence of solutions, it is fundamental that we control the high norms of the solutions, especially the supnorm (see, e.g., 77. O. Ladyzhenskaya, V.A. Solonnikov & N.N. Uralceva. “Linear and Quasilinear Equations of Parabolic Type”. American Mathematical Society, Providence (1968).). In general situations, the task of controlling can become very difficult. To illustrate this question, intuitively, let us consider positive solutions of the following equation, simpler than that of the problem (1.1): , that we can rewrite as . In this equation, the dissipative term tends to make the magnitude of the solution decrease, but the term , in regions where is negative, tends to make the magnitude of the solution increase. The result of this competition isn’t easy to predict.
In 33. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 6, v.2. SBMAC (2018). doi:10.5540/03.2018.006.02.0273.
https://doi.org/10.5540/03.2018.006.02.0...
we considered more general conditions for the function and obtain for solutions of (1.1), for all , the following supnorm estimate:
with e , where , for , and for . This controls the supnorm, but it is does not guarantee the existence of global solutions to (1.1), for . In this work, we will consider and we will present conditions over κ with which the existence of global solution is guaranteed.
The equation in (1.1) generalizes several important equations, including the Porous Media Equation and the p-Laplacian Equation ( and ); and is considered, for example, in 44. E. DiBenedetto. “Degenerate Parabolic Equations”. Springer-Verlag, New York (1993). and in 99. Z. Wu, J. Zhao, J. Yin & H. Li. “Nonlinear Diffusion Equations”. World Scientific, Singapore (2001).. The basic ideas used in our procedure can be seen in 55. P.B. e Silva, L. Schütz & P. Zingano. On some energy inequalities and supnorm estimates for advection- diffusion equations in Rn. Nonlinear Analysis: Theory, Methods & Applications, 93 (2013), 90-96. doi:10.1016/j.na.2013.07.028.
https://doi.org/10.1016/j.na.2013.07.028...
, where they are apllied to a somewhat simpler equation and where the additional condition (1.3) is satisfied; and the ideas presented in this paper can be seen in more detail in 11. J.Q. Chagas. “Contribuições para a teoria de equações parabólicas duplamente não lineares com termos advectivos”. Ph.D. thesis, UFRGS, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (2015)..
2 HYPOTHESES AND PRELIMINARY RESULTS
In this Section we rank the preliminary results to be used in the proof of the major result of this paper: the Theorem 2.3, presented in Section 3.
We begin presenting some reasonable hypothesis about the function and about the solution of (1.1), for and , that we use to obtain our results.
Consider , where is a smooth function satisfying , such that satisfies (1.2). denotes the variation in in ℝn: for each we become B i (t) by
and therefore for each . Besides, we require that satisfies .
As the solution to (1.1) is limited for each , we consider the following limitation estimate: . We also assume the convergence of the solution to the given initial value u 0 as , is in the sense of . In addition, we suppose that for each and, therefore, for each we have an estimate in the form , for all .
The first result in this section is specific for in (1.1): it is verified that, for in the maximal interval of existence, the solutions have decreasing , more specifically:
Theorem 2.1.Letbe a solution for (1.1) with, for. Then,
In particular, when,
As well as the norm decrease, also the properties of contraction in , mass conservation and comparison principle are valid. The proofs of these properties, including Theorem 2.1, can be found in 11. J.Q. Chagas. “Contribuições para a teoria de equações parabólicas duplamente não lineares com termos advectivos”. Ph.D. thesis, UFRGS, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (2015)., but there’s no novelty in these properties as they are already known and plenty of proofs can be found in the literature.
In 33. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 6, v.2. SBMAC (2018). doi:10.5540/03.2018.006.02.0273.
https://doi.org/10.5540/03.2018.006.02.0...
we present the estimate (1.4) for the norm of the sup for the solutions of (1.1), valid for , and the proof is outlined in the following. We begin with an important energy inequality:
Theorem 2.2.Letsolution of (1.1) for. Supposefor some, then
for all q satisfyingand, and for all, whereis a null set.
Proof. See 33. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 6, v.2. SBMAC (2018). doi:10.5540/03.2018.006.02.0273.
https://doi.org/10.5540/03.2018.006.02.0...
.
Then, for , the hypothesis is changed to , where σ satisfies and , with γ− denoting the negative part of must satisfy, additionaly, the condition . Together with the energy inequality (2.2), we use the interpolation inequality of Sobolev-Nirenberg-Gagliardo (SNG) type:
where , and r, s and satisfies , and (for more details about this inequality, see, for instance, 66. A. Friedman. “Partial Differential Equations”. Holt, Rinehart and Winston, New York (1969).), to prove what we call Fundamental Lemma, a result relating the norms L q and L q/σ of the solutions :
Lemma 2.1. (Fundamental Lemma)Letsolution of (1.1) for. If q satisfiesand, then, for each,
whereand.
Proof.: See 33. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 6, v.2. SBMAC (2018). doi:10.5540/03.2018.006.02.0273.
https://doi.org/10.5540/03.2018.006.02.0...
.
The Fundamental Lemma is used in an iterative process to obtain, after a few steps, the estimate (1.4) for the limitation of the norm of the sup of the solution of (1.1), for :
Theorem 2.3.Letsolution of (1.1) for. Given, for each,
whereand.
Proof. In the first step we use the iterative argument that allows to estimate the norms L q of the solution for large values of q, in the interval , with , as a function of lower norms of u: we show that, for solution to (1.1), given , for each ,
where . Besides, for each ,
with . The proof of (2.6) is done by induction, taking in (2.4); then taking in (2.4), and likewise thereafter. Through simple manipulations of exponents, we obtain that, for each
for . Besides, defining , we rewrite (2.7), for each , as
Next, using the inequality of norms interpolation , where and satisfy , we obtain a simpler estimate for by estimating the intermediate terms of (2.8) appropriately and putting toghether with the first and last terms: we prove that, given , for each ,
where .
Simply by letting in (2.9), we obtain the following estimate for the limitation of the norm of the sup of the solution for (1.1): given , for each ,
Finaly, to obtain (2.5), we only recall the definition of , with , and define δ 1 and δ 2, respectively, as:
3 MAIN RESULT
In this section we apply the results listed in Section 2 to obtain conditions for the global existence (i.e., conditions that guarantee that ) of solutions of (1.1), with , i.e.,
The first step is to obtain an energy inequality in an adequate form for the application.
Lemma 3.2.Letbe a solution to (3.1) for. Supposefor some,
for all q satisfyingand, and for all, whereis a null set.
Proof. We start with the energy inequality presented in (2.2). By taking an upper limit to the right-hand side and using Ho¨lder, we obtain
where .
Defining by , for all , where , we rewrite (3.3) as
where λ, e λ0 are, respectivaly, ; and , with σ satisfying and .
Then, using the inequality of the type SNG in (2.3), with and , we rewrite the inequality in (3.4) as
where θ and are giving by
After using (3.5) in (3.4) and substituting θ and by their respective values, we obtain
Finally, rewriting (3.6) in terms of u, we arrive at
In the following we present the major result of this work: the determination of constraints on that guarantee the existence of global solutions to (3.1).
Theorem 3.4. Under the hypothesis of Theorems 2.1 and 2.3, the solutions to (3.1) satisfy:
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(i)If, thenis defined for all(given any initial u0).
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(ii)If, the solutions are global when
where C is the constant in the SNG inequality used to obtain (3.2).
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(iii)If, the solutions are global when the initial u0satisfies
where
Proof. Case (i):.
By Theorem 2.3, taking , we obtain
By Theorem 2.1, we have that . Then, it follows that
for all . Therefore, the solution can be extended past T ∗, and we conclude that solutions are limited for all , i.e., the solution is global. This concludes case (i).
Case (ii): .
In this case, we have
and, therefore, the resctriction does not allow the use of Theorem 2.3 with , so that we cannot use the decrease of the norm L 1 of the solution, as in case (i).
For κ in this case, we have
and, therefore, , so that the conditions about σ are satisfied for all . We choose , and, consequently, the restriction makes it so that if is to be satisfied, then . For simplicity, we use . With these choices, we use (3.2) to estimate . This is the norm we will be using in Theorem 2.3 to control in this case. Rewriting (3.2), using and , substituting , and taking an upper limit to B(t) on (0, t), we arrive at
Note that (3.10) informs that for all , provided that
i.e., for some . In this cenario, then is decreasing in (0, t), and therefore the solution to (3.1) is defined in (0, t). In particular, if (3.7) is satisfied, i.e.,
then the solution to (3.1) is globally defined (i.e., ). This concludes case (ii).
Case (iii): .
In this case, we choose
Again, it is not possible to apply Theorem 2.3 directly because does not satisfy the condition . Besides, with the values of κ for this case, we have that , meaning we cannot use the decrease of norm L 1 of the solution. As in case (ii), the values for κ in this case lead us to , therefore, the conditions on σ are satisfied by any . For simplicity, we use again .
With these choices for and σ, we try to use Theorem 2.3 to control the norm of the sup of the solution by its L q norm, with .
Rewriting the inequality (3.2) using the chosen values for and σ , substituting γ and taking the sup on (0, t) of B(t) as an upper limit to it, we obtain
As in case (ii), (3.12) would give us , provided that
i.e.,
where .
The condition (3.13) is sufficient to guarantee that is decreasing (and therefore controls, by Theorem2.3, the sup norm). The problem is, up to this moment, we do not control .
To find out what is necessary so that (3.13) is satisfied, we proceed as follows: we use the interpolation of norms inequatily to obtain
we raise both sides of (3.14) to power, and multiply by , with as in (3.9), to obtain
Then we use again the interpolation of norms inequality to estimate , i.e.,
and, substituting (3.16) in (3.15), we obtain
We claim that, if we make satisfy the condition (3.8), rewritten as
where is given in (3.9) and in (3.11), we will obtain
at least for close to zero. But this is sufficient to guarantee as well that (3.18) is valid for any . Indeed, if (3.18) was not true for any , then there would exists T 1, with , for which
However, if (3.19) is valid, by (3.17) it follows that the inequality in (3.13) is valid for any and, therefore, (3.12) implies that is decreasing, for all . Then, by Theorem 2.3, controls the norm of the sup, for all . As we have let u 0 satisfy (3.8), and both the norm L 1 and the norm L ∞ decrease in (0, T 1) (the norm L ∞ decreases in (0, T 1) by (3.19) and by the decrease of the norm L 1), by the continuity of the solution, if is not possible that we have (3.20).
Therefore, the inequality (3.18) is valid for all , and then (3.13) is also satisfied for all . We conclude by (3.12) that decreases monotonically.
We then conclude, by Theorem 2.3, that the solution exists for all , provided that (3.8) is satisfied.
This ends the proof of Theorem 3.4. □
4 CONCLUSION
In this work we consider the initial-value problem (1.1) for an doubly nonlinear advection-diffusion equation and, with , we presented conditions over κ in (1.2) with which the existence of global solutions is guaranteed. In the general case where the sup norm is controlled in a specific form (see 22. J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: um caso de decrescimento. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 5, v.1. SBMAC (2017). doi:10.5540/03.2017.005.01.0034.
https://doi.org/10.5540/03.2017.005.01.0...
), as an application of Theorems 2.1 and 2.3, in our main result, Theorem 3.2 (Section 3), we guaranteed that the condition over κ for which the global existence of solution is always obtained is , i.e, independent of the initial profile u
0, for these valors of κ the solutions are globals . Even if or , we presented conditions over u
0 for which the global solutions yet are possible.
ACKNOWLEDGEMENTS
This work was partly supported by CNPq, Brazil.
REFERENCES
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1J.Q. Chagas. “Contribuições para a teoria de equações parabólicas duplamente não lineares com termos advectivos”. Ph.D. thesis, UFRGS, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (2015).
-
2J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: um caso de decrescimento. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 5, v.1. SBMAC (2017). doi:10.5540/03.2017.005.01.0034.
» https://doi.org/10.5540/03.2017.005.01.0034 -
3J.Q. Chagas, P.L. Guidolin & P.R. Zingano. Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral. In “Proceeding Series of the Brazilian Society of Applied and Computational Mathematics”, 6, v.2. SBMAC (2018). doi:10.5540/03.2018.006.02.0273.
» https://doi.org/10.5540/03.2018.006.02.0273 -
4E. DiBenedetto. “Degenerate Parabolic Equations”. Springer-Verlag, New York (1993).
-
5P.B. e Silva, L. Schütz & P. Zingano. On some energy inequalities and supnorm estimates for advection- diffusion equations in Rn. Nonlinear Analysis: Theory, Methods & Applications, 93 (2013), 90-96. doi:10.1016/j.na.2013.07.028.
» https://doi.org/10.1016/j.na.2013.07.028 -
6A. Friedman. “Partial Differential Equations”. Holt, Rinehart and Winston, New York (1969).
-
7O. Ladyzhenskaya, V.A. Solonnikov & N.N. Uralceva. “Linear and Quasilinear Equations of Parabolic Type”. American Mathematical Society, Providence (1968).
-
8D. Serre. “Systems of Conservation Laws”, volume 1. Cambridge University Press, Cambridge (1999).
-
9Z. Wu, J. Zhao, J. Yin & H. Li. “Nonlinear Diffusion Equations”. World Scientific, Singapore (2001).
Publication Dates
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Publication in this collection
30 Apr 2020 -
Date of issue
Jan-Apr 2020
History
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Received
10 Dec 2018 -
Accepted
27 Sept 2019