Abstract
We establish asymptotic stability estimates for solutions to evolution problems with singular convection term. Such quantitative estimates provide a measure with respect to the time variable of the distance between the solution to a parabolic problem from the one of the its elliptic stationary counterpart.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
This paper concerns evolution problems whose model case reads as follows
Here and in what follows \(\Omega \) denotes a regular bounded domain of \({\mathbb {R}}^N\) with \(N\geqslant 3\), \(A>0\), \( T\in (0,\infty ]\) and \(\Omega _T\) stands for the cylinder \(\Omega \times (0,T) \). With regard to the structure assumptions of the problem, we assume that \(M=M(x,t):\Omega \times (0,T)\rightarrow {\mathbb {R}}^{N\times N}\) is a measurable, symmetric, matrix field satisfying the uniform bounds
for every \(\xi \in {\mathbb {R}}^N\) and for a.e. \((x,t) \in \Omega \times (0,T)\) where \(0<\lambda \leqslant \Lambda \). For the data of the problem we assume that
The aim of this note is to provide a quantitative estimate related to the long time behavior of the global in time weak solution \(u=u(x,t)\) of (1.1) (according to Definition 3.1 below). As an example, we wonder whether the solution \(u=u(x,t)\) defined in the whole of \(\Omega _\infty \) tends toward the one of the stationary problem
as \(t\rightarrow \infty \). For the data and for the structure assumptions relative to problem (1.4), we assume
If all the assumptions above are fulfilled, an important property for the elliptic problem (1.4) relies on the fact that that if a solution exists then it is automatically unique (see e.g. [16]). Obseve that our problem exhibits an unbounded and singular convection term if \(0\in \Omega \), because of the presence of coefficient \(E_A(x):= A \frac{x}{|x|^2} \).
We introduce the following functions
which can be read as measures in time of the distances between the matrix M and the identity \(\mathbf{I}\) and F and f respectively. We assume that
Finally, we set
We assume that \(\Omega \) contains the origin (so that the coefficient \(E_A\) is singular) and we state our result related to problem (1.1).
Theorem 1.1
Assume that the solutions to problems (1.1) and (1.4) exist. If \(0 \in \Omega \) and if
then
for some positive constants \(\mu \) and \(C_0\). Moreover, if \(T=\infty \) and \( {\mathcal {K}} \in L^1([t_0,\infty ))\) then
In the latter case, we have
Let us spend few comments on condition (1.8). First, one can observe that the time independent coefficient \(E_A(x):= A \frac{x}{|x|^2} \) appearing in (1.1) actually belongs to the Marcinkiewicz space \(L^{N,\infty } (\Omega )\) (we refer the reader to Sect. 2.1 for the definition and the basic properties of this function space) but does not belong to \(L^{N} (\Omega )\) as long as \(0\in \Omega \). Moreover
where \(\omega _N\) stands for the measure of the unit ball in \(\mathbb R^N\). In (1.11) the distance from \(L^\infty \) in \(L^{N,\infty }\) appears, as defined in Sect. 2.1 below. With regard to general problems of the type
the results of [9] state that one cannot expect existence of a solution (according again to Definition 3.1 below) unless we assume some uniform with respect to the time variable bound on the distance of the convective field E from \(L^\infty \) in \(L^{N,\infty }\). Therefore, condition (1.8) seems quite natural in our framework in light of (1.11).
Comparison quantitative estimates between solutions of evolutionary and stationary problems as in (1.9) or (1.10) (see also (3.19) or (3.20) below) are available in [17] for equations not having lower order terms. It should be also worth to mention that recently in [10] new estimates for the behaviour at infinity of solutions to a wide class of parabolic partial differential equations (including also anisotropic type equations) have been considered.
Among all possible equations taking a form as in (1.1) we mention the following homogeneous one
which is known as Fokker–Planck equation. Its relevance in literature depends upon the fact that such equation describes the evolution of some Brownian motion and of some Mean Field Game. In case the convective term is bounded, many results are available in literature (see e.g. [5] and the references therein). On the other hand, in some context (see e.g. the case of the diffusion model for semiconductor devices in [6]) the boundedness of the convective field is not immediately guarantee. In addition to Definition 3.1 below, further definitions of solutions have been introduced for problem (1.1) under consideration as the renormalized solution (see e.g. [19] where the Fokker–Planck equation is coupled with some Hamilton–Jacobi–Bellman equation) and entropy solutions (see e.g. [3] where the authors do not address the existence of weak solution and obtain the existence of entropy solution assuming that \(E\in L^2(\Omega _T,{\mathbb {R}}^N)\)).
The plan of this paper is the following. In Sect. 2 we introduce the function spaces which are related to our problems and some useful results which help us in proving the asymptotic behaviour of Theorem 1.1. In Sect. 3 we will actually prove a result for a general Cauchy–Dirichlet problem, in such a way that Theorem 1.1 is a special case of this statement. The presence of the lower order term does not allow to follow [17]. We establish an estimate of decay of the super-level sets of the solution which is fundamental in order to obtain our result. Nevertheless, at the end of Sect. 3 we will underline how the assumption (1.8) comes into play for the special problem (1.1).
2 Preliminary results
2.1 Lorentz spaces
Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^N\). From now on the Lebesgue measure of a measurable subset E of \({\mathbb {R}}^N\) will be denoted by |E|. Fixed \(p,q \in (1,\infty )\), the Lorentz space \(L^{p,q} (\Omega )\) corresponds to the class of all measurable functions g defined on \(\Omega \) for which the quantity
is finite, where \(\Omega _\tau = \{x\in \Omega :\, |g(x)|>\tau \}\) for any \(\tau >0\). A standard feature of \(\Vert \cdot \Vert _{p,q} \) relies on the fact that it is equivalent to a norm with the property that \(L^{p,q}(\Omega )\) becomes a Banach space when endowed with it (we refer the reader to [18]). When \(p=q\), the Lorentz space \(L^{p,p}(\Omega )\) reduces to the classical Lebesgue space \(L^{ p}(\Omega )\). On the other hand, when \(q=\infty \), the class \(L^{p,\infty }(\Omega )\) corresponds to the class of all measurable functions g defined on \(\Omega \) for which the quantity
is finite. The class \(L^{p,\infty }(\Omega )\) is known as the Marcinkiewicz class and it is usually also denoted by \(\text {weak}-L^p\). Moreover, if we set
it results
We refer the reader to Lemma A.2 in [2] for the proof of the latter relation.
For the Lorentz spaces the following inclusions hold
whenever \(1\leqslant q< p < r \leqslant \infty \). Moreover, for \(1< p < \infty \), \(1\leqslant q \leqslant \infty \) and \( \frac{1}{p} + \frac{1}{p^\prime } =1 \), \( \frac{1}{q} + \frac{1}{q^\prime } =1 \), if \(f \in L^{p,q}(\Omega )\) and \(g \in L^{p^\prime ,q^\prime }(\Omega )\), we have the Hölder–type inequality
It is well known that \(L^\infty (\Omega )\) is not a dense subspace of \(L^{p,\infty }(\Omega )\). The distance to \(L^\infty (\Omega )\) in \(L^{p,\infty }(\Omega )\) is defined as
We conclude this Section by recalling the Sobolev embedding theorem in the setting of Lorentz spaces in the sharp form given by [1].
Theorem 2.1
Let us assume that \(1<p<N\) and \(1 \leqslant q \leqslant p\). If \( u \in W^{1,1}_0 (\Omega )\) is a function whose gradient satisfies \(| \nabla u | \in L^{p,q}(\Omega )\) then \(u\in L^{p^*,q}(\Omega )\) where \(p^*=\frac{Np}{N-p}\) is the usual Sobolev exponent and
where \(S_{N,p} = \omega _N^{-1/N} \frac{p}{N-p} \) and \(\omega _N\) is the measure of the unit ball in \({\mathbb {R}}^N\).
2.2 Suitable subsets of the space \(L^\infty \left( 0,T ; L^{p,\infty } (\Omega ) \right) \)
Given \(T\in (0,\infty ]\) and \(\delta \geqslant 0\), we consider the subset \(X_{\delta } (\Omega _T)\) of \(L^\infty \left( 0,T ; L^{p,\infty } (\Omega ) \right) \) defined as
In other words, \(X_\delta (\Omega _T)\) consists of of all those functions \(f \in L^\infty \left( 0,T ; L^{p,\infty } (\Omega ) \right) \) such that there exists \( g \in L^\infty \left( 0,T ; L^{ \infty } (\Omega ) \right) \) such that
Clearly \( X_0 (\Omega _T) \) is the closure of \( L^\infty \left( \Omega _T\right) \) in \( L^\infty \left( 0,T ; L^{p,\infty } (\Omega ) \right) \) and
for \(p\leqslant q < \infty \).
A characterization of \(X(\Omega _T)\) can be given in terms of the the truncation operator at level \(\pm \kappa \) (for \(\kappa >0\)), that is
for \(s\in {\mathbb {R}}\). The following lemma then follows (see e.g. [9]).
Lemma 2.2
For any given \(\delta \geqslant 0\), \( f\in X_\delta (\Omega _T)\) if and only if
2.3 Abstract asymptotic estimates
An essential tool in the study of the time behaviour of our problem relies on the following result, whose proof can be found in [17].
Proposition 2.3
Let \(t_0 \geqslant 0\) and \(T \in (t_0 , +\infty ]\). Assume that \(\phi =\phi (t)\) is a continuous and non negative function defined in \([t_0,T)\) verifying
for every \(t_0 \leqslant t_1< t_2 <T\), where M is a positive constant and g is a non negative function belonging to \(L^1([t_0,T))\). Then, for every \(t \geqslant t_0\) we get
Moreover, if \(T=+\infty \) and g belongs to \(L^1([t_0,+\infty ))\) there exists \(t_1 > t_0\) such that
for every \(t \geqslant t_1\), where
.
3 Existence and uniqueness to the a more general parabolic problem
In this Section we consider the following evolution problem
which turns to be more general than the one in (1.1), because of the structure assumptions that we are going to describe below. Once again, \(\Omega \) is a regular bounded domain of \({\mathbb {R}}^N\) with \(N\geqslant 3\), \( T\in (0,\infty ]\) and \(\Omega _T\) stands for the cylinder \(\Omega \times (0,T) \). For the data of the problem we assume that (1.3) holds true. The vector field \(A=A(x,t,\xi ) :\Omega _T \times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) is a Carathéodory function satisfying the following conditions
for a.e. \((x,t)\in \Omega _T\) and for any \(\xi ,\eta \in {\mathbb {R}}^N\). Moreover, we assume that \(B=B(x,t,s) :\Omega _T \times \mathbb R \rightarrow {\mathbb {R}}^N\) is a Carathéodory function satisfying the following properties
for a.e. \( x \in \Omega \), for any \(t \in (0,T)\), for any \(s,s^\prime \in {\mathbb {R}}\) and for some suitable measurable function \(b:\Omega _T \rightarrow [0,\infty )\). With a slight abuse of terminology, the function b in (3.4) is called convective term. Concerning the regularity of the convective term, we will assume from now on that
We consider weak solutions of our problem, according to the following definition.
Definition 3.1
We say that
is a weak solution to problem (3.1) if one has
for all \(\varphi \in C^\infty (\Omega _T)\) with \({{\mathrm{supp}}} \, \varphi \subset \subset \Omega \times [0,T)\). We say that u is a global weak solution if
and (3.7) holds true for any given \(T>0\) with \(\varphi \) as before.
The main goal of the present section is to introduce suitable conditions allowing that the solution of (3.1) tends as \(t\rightarrow \infty \) toward the one of the stationary problem
For the data relative to problem (3.8) and for the structure assumptions on the Carathéodory functions \(\tilde{A}={{\tilde{A}}}(x ,\xi ) :\Omega \times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) and \({{\tilde{B}}}={{\tilde{A}}}(x ,s) :\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}^N\), we require that
for some \(0<\alpha ^\prime \leqslant \beta ^\prime <\infty \). Problem (3.1) and its stationary counterpart (3.8) have a common feature as far as existence and uniqueness of a solution are concerned. Indeed, problem (3.8) admits a unique weak solution in \(W^{1,2}_0(\Omega )\) if
for some \(\delta \geqslant 0\) depending on the structure assumption of the problem and on N (see e.g. [11, 12]). It is worth mentioning that existence of a solution to (3.8) could possibly fail if \(\delta \) in (3.13) is too large (see Section 4 in [12]). We also recall that in the elliptic framework a condition like (1.8) guarantees existence results (see e.g. [4]). We stress that (3.13) and (1.8) can be compared as done in the Introduction. Similarly, existence and uniqueness for problem (3.1) is obtained by assuming that the convective term \(b=b(x,t)\) satisfies (3.13) uniformly with respect to the time variable, i.e.
for some small \(\delta >0\) depending only on \(\alpha \) and N. As before, we introduce the following subset of \({L^\infty \left( 0,T; L^{N,\infty } (\Omega ) \right) }\)
Let us assume from now on that some \(t_0\geqslant 0\) exists such that
holds true for \(t \geqslant t_0\) and where \(G_0,G\) and H belongs to \(L^1([t_0,\infty ))\). According to the terminology of [17], we refer to (3.16), (3.17) and (3.18) as proximity conditions.
The main result of this section reads as follows.
Theorem 3.1
Assume that both problems (3.1) and (3.8) admit solution. There exists \(\delta >0\) depending only on \(\alpha \) and N such that, if \(b \in X_\delta (\Omega _T)\) then
for some positive constants \(\mu \) and \(C_0\) and where
Moreover, if u is a global solution of (3.1) and if \(K \in L^1([t_0,\infty ))\) then
We remark that stability and continuity estimates similar to the ones of Theorem 3.1 are also available in [7,8,9, 17]. On the other hand, it is also worth to mention that also pointwise estimates of the spatial gradient are available in literature (see e.g. [14, 15]).
4 Proof of the main result
Before we give the proof of Theorem 3.1, we focus our attention to a technical result of its own interest which describes the decay of the measure of the superlevel sets of the difference of the solutions of (3.1) and (3.8). Indeed, in the spirit of [4, 9, 16] we prove the following lemma.
Lemma 4.1
Assume that the solutions to problems (3.1) and (3.8) exist. Fixed \(T\in (0,\infty ]\) and \(k>0\) we have
for some constant \(C>0\) which only depends on \(N, \alpha \), \(\Vert \nabla v\Vert _{L^2(\Omega )}\), \(\Vert G\Vert _{L^1(0,T)}\), \(\Vert H\Vert _{L^1(0,T)}\), \(\Vert G_0\Vert _{L^1(0,T) }\) and \( \Vert b\Vert _{L^2(\Omega _T)}\).
Proof
We fix \(T^*\in (0,T)\), we test both equations in problems (3.1) and (3.8) with \(\varphi =\frac{w}{1+|w|}\chi _{(0,T^*)}\) where \(w:=u-v\) and we subtract in order to get
We observe that \(\nabla \varphi = \frac{\nabla w}{(1+|w|)^2} \chi _{(0,T^*)}\). The first term at the left hand side of (4.2) can be estimated as follows
so (4.2) is equivalent to
We observe that
and, since \(e^\sigma - 1 - \sigma \geqslant \frac{1}{2} \sigma ^2\) for any \(\sigma \geqslant 0\), we have
So we have
From the monotonicity we have
From (4.2) and observing that \(w(\cdot ,0)=u_0-v\) we get
Because of Young’s inequality, for \(\delta >0\) we have
and also
Taking into account all the above estimates, the fact that \(|\nabla \varphi |^2 \leqslant \frac{|\nabla w|^2}{(1+|w|)^2}\chi _{(0,T^*)}\) and (4.4), we have
We choose \(\delta =\alpha /3\) and reabsorb at the left hand side to get
Since \(T^*\) can be arbitrarily chosen in (0, T) and recalling conditions (3.16), (3.17) and (3.18), we immediately obtain the desired conclusion. \(\square \)
Proof of Theorem 3.1
Let us require that
where \(S_{N,2}\) is the sharp Sobolev constant appearing in (2.2) whenever \(p=2\). This means that
and so there exists \(M>0\) such that
For fixed \(t_1,t_2 \in (t_0,\infty )\) with \(t_1<t_2\), we use \(\varphi =(u-v) \chi _{(t_1,t_2)}\) as a test function in both (3.1) and (3.8) and subtract the results obtained from this method to get
We have rearranged the terms in (4.12) so that we may argue as in the proof of Lemma 4.1 to obtain
The only issue that matters is to estimate the latter term in (4.13). We observe that
We set \(\omega _k(t) := |\{x\in \Omega :\, |u(x,t)-v(x)| >k | \). We use (2.1), Sobolev inequality (2.2) and (4.11) to get
On the other hand, using again (2.1) and (2.2)
Due to Lemma 4.1, we have \(\omega _k(t) \rightarrow 0\) as \(k\rightarrow \infty \) uniformly w.r.t. \(t\in (0,T)\). So, if we choose k sufficiently large to have
Then, using Young’s inequality we have
Inserting (4.18) in (4.15) we have
Taking into account (4.13), we reabsorb by the left hand side and then we may find some constants \(c_1\) and \(c_2\) such that
where \(w:=u-v\). Finally, by means of Poincaré inequality, we have
Thus, we may apply Proposition 2.3 to the function
and estimate (4.21) immediately yelds the desired result. \(\square \)
Proof of Theorem 1.1
With respect to the proof of previous Theorem 3.1, we perform a different argument to estimate of the term
appearing in (4.12). Indeed, in this case we have \(B(x,t, u)=A\frac{x}{|x|^2}u\) and so the latter term can be estimated
Now, we make use of the classical Hardy inequality in its sharp form (see e.g. Lemma 17.1 in [20] for an elementary proof)
to get
The latter term can be reabsorbed by the left hand side because of condition (1.8), while the rest of the proof goes without changes. \(\square \)
References
Alvino, A.: Sulla disuguaglianza di Sobolev in spazi di Lorentz. Bollettino della Unione Matematica Italiana A 5(14), 148–156 (1977)
Bénilan, P., Brezis, H., Crandall, M.: A semilinear equation in \(L^1({\mathbb{R}}^N)\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(4), 523–555 (1975)
Boccardo, L., Orsina, L., Porretta, A.: Some noncoercive parabolic equations with lower order terms in divergence form, Dedicated to Philippe Bénilan. J. Evol. Equ. 3(3), 407–418 (2003)
Boccardo, L.: Dirichlet problems with singular convection terms and applications. J. Differ. Equ. 258(7), 2290–2314 (2015)
Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games. Netw. Heterog. Media 7(2), 279–301 (2012)
Fang, W., Ito, K.: Weak solutions for diffusion–convection equations. Appl. Math. Lett. 13(3), 69–75 (2000)
Farroni, F., Greco, L., Moscariello, G.: Stability for \(p\)-Laplace type equation in a borderline case. Nonlinear Anal. 116, 100–111 (2015)
Farroni, F., Greco, L., Moscariello, G.: Estimates for \(p\)-Laplace type equation in a limiting case. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25(4), 445–448 (2014)
Farroni, F., Moscariello, G.: A nonlinear parabolic equation with drift term. Nonlinear Anal. B 177, 397–412 (2018)
Frankowska, H., Moscariello, G.: Long-Time behaviour of solutions to an evolution PDE with nonstandard growth. Adv. Calc. Variat. (2020). https://doi.org/10.1515/acv-2019-0061
Giannetti, F., Greco, L., Moscariello, G.: Linear elliptic equations with lower-order terms. Differ. Integ. Equ. 26(5–6), 623–638 (2013)
Greco, L., Moscariello, G., Zecca, G.: Very weak solutions to elliptic equations with singular convection term. J. Math. Anal. Appl. 457(2), 1376–1387 (2018)
Greco, L., Moscariello, G., Zecca, G.: An obstacle problem for noncoercive operators. Abst. Appl. Anal. Art. ID 890289 (2015)
Kuusi, T., Mingione, G.: The Wolff gradient bound for degenerate parabolic equations. J. Eur. Math. Soc. (JEMS) 16(4), 835–892 (2014)
Kuusi, T., Mingione, G.: Riesz potentials and nonlinear parabolic equations. Arch. Ration. Mech. Anal. 212(3), 727–780 (2014)
Moscariello, G.: Existence and uniqueness for elliptic equations with lower-order terms. Adv. Calc. Var. 4(4), 421–444 (2011)
Moscariello, G., Porzio, M.M.: Quantitative asymptotic estimates for evolution problems. Nonlinear Anal. 154, 225–240 (2017)
O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)
Porretta, A.: Weak solutions to Fokker-Planck equations and mean field games. Arch. Ration. Mech. Anal. 216(1), 1–62 (2015)
Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Lect. Notes Unione Mat. Italiana 3. Springer , Berlin(2007)
Funding
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The Author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Farroni, F. Asymptotic stability estimates for some evolution problems with singular convection field. Ricerche mat 71, 441–457 (2022). https://doi.org/10.1007/s11587-020-00537-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-020-00537-1