Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $ddot{u}(t) + Au(t) + g(dot{u}(t))=h(t),quad tinmathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.
{"title":"On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator","authors":"A. Haraux","doi":"10.2140/tunis.2019.1.59","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.59","url":null,"abstract":"Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $ddot{u}(t) + Au(t) + g(dot{u}(t))=h(t),quad tinmathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.59","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45191818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.
{"title":"Diffusion-approximation in stochastically forced kinetic equations","authors":"A. Debussche, J. Vovelle","doi":"10.2140/TUNIS.2021.3.1","DOIUrl":"https://doi.org/10.2140/TUNIS.2021.3.1","url":null,"abstract":"We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2021.3.1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47373915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-19DOI: 10.2140/tunis.2019.1.151
P. Auscher, S. Bortz, Moritz Egert, Olli Saari
A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi–Mingione–Sire and Bass–Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity.
{"title":"Nonlocal self-improving properties: a functional analytic approach","authors":"P. Auscher, S. Bortz, Moritz Egert, Olli Saari","doi":"10.2140/tunis.2019.1.151","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.151","url":null,"abstract":"A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi–Mingione–Sire and Bass–Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.151","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47045074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the semilinear heat equation begin{eqnarray*} partial_t u = Delta u + |u|^{p-1} u ln ^{alpha}( u^2 +2), end{eqnarray*} in the whole space $mathbb{R}^n$, where $p > 1$ and $ alpha in mathbb{R}$. Unlike the standard case $alpha = 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: begin{eqnarray*} u(x,t) sim psi(t) left(1 + frac{(p-1)|x-a|^2}{4p(T -t)|ln(T -t)|} right)^{-frac{1}{p-1}} text{ as } t to T, end{eqnarray*} where $psi(t)$ is the unique positive solution of the ODE begin{eqnarray*} psi' = psi^p ln^{alpha}(psi^2 +2), quad lim_{tto T}psi(t) = + infty. end{eqnarray*} The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.
{"title":"Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation","authors":"G. K. Duong, V. T. Nguyen, H. Zaag","doi":"10.2140/tunis.2019.1.13","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.13","url":null,"abstract":"We consider the semilinear heat equation begin{eqnarray*} partial_t u = Delta u + |u|^{p-1} u ln ^{alpha}( u^2 +2), end{eqnarray*} in the whole space $mathbb{R}^n$, where $p > 1$ and $ alpha in mathbb{R}$. Unlike the standard case $alpha = 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: begin{eqnarray*} u(x,t) sim psi(t) left(1 + frac{(p-1)|x-a|^2}{4p(T -t)|ln(T -t)|} right)^{-frac{1}{p-1}} text{ as } t to T, end{eqnarray*} where $psi(t)$ is the unique positive solution of the ODE begin{eqnarray*} psi' = psi^p ln^{alpha}(psi^2 +2), quad lim_{tto T}psi(t) = + infty. end{eqnarray*} The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.13","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45901813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-03-01DOI: 10.2140/tunis.2020.2.287
Bruce W. Jordan, Z. Klagsbrun, B. Poonen, C. Skinner, Yevgeny Zaytman
For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ is as predicted by this conjecture.
{"title":"Statistics of K-groups modulo p for the ring of\u0000integers of a varying quadratic number field","authors":"Bruce W. Jordan, Z. Klagsbrun, B. Poonen, C. Skinner, Yevgeny Zaytman","doi":"10.2140/tunis.2020.2.287","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.287","url":null,"abstract":"For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ is as predicted by this conjecture.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.287","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48171694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-02-15DOI: 10.2140/tunis.2019.1.539
H. Henn
Let Γ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic Γ-CW complex on which Γ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel constructon of Xs with respect to the action of Γ. This cohomology coincides with the mod-2 cohomology of Γ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the strucure of the cohomology ring H * (Γ; Z/2).
{"title":"On the mod-2 cohomology of\u0000SL3ℤ,i","authors":"H. Henn","doi":"10.2140/tunis.2019.1.539","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.539","url":null,"abstract":"Let Γ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic Γ-CW complex on which Γ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel constructon of Xs with respect to the action of Γ. This cohomology coincides with the mod-2 cohomology of Γ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the strucure of the cohomology ring H * (Γ; Z/2).","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.539","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43529948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-23DOI: 10.2140/tunis.2019.1.221
Z. Ammari, S. Breteaux, F. Nier
We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.
{"title":"Quantum mean-field asymptotics and multiscale analysis","authors":"Z. Ammari, S. Breteaux, F. Nier","doi":"10.2140/tunis.2019.1.221","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.221","url":null,"abstract":"We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.221","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45672966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish the vanishing of the third unramified cohomology group for many types of Fano hypersurfaces in projective space over an algebraically closed field of arbitrary characteristic, and over a finite field. For cubic hypersurfaces over a finite field, the case of fourfolds remains open. --- Sur un corps alg'ebriquement clos et sur un corps fini, on 'etablit de nouveaux r'esultats d'annulation pour la cohomologie non ramifi'ee de degr'e 3 pour de nombreux types d'hypersurfaces de Fano. Le cas des hypersurfaces cubiques de dimension 4 sur un corps fini reste ouvert.
{"title":"Troisième groupe de cohomologie non\u0000ramifiée des hypersurfaces de Fano","authors":"Jean-Louis Colliot-Th'elene","doi":"10.2140/TUNIS.2019.1.47","DOIUrl":"https://doi.org/10.2140/TUNIS.2019.1.47","url":null,"abstract":"We establish the vanishing of the third unramified cohomology group for many types of Fano hypersurfaces in projective space over an algebraically closed field of arbitrary characteristic, and over a finite field. For cubic hypersurfaces over a finite field, the case of fourfolds remains open. \u0000--- \u0000Sur un corps alg'ebriquement clos et sur un corps fini, on 'etablit de nouveaux r'esultats d'annulation pour la cohomologie non ramifi'ee de degr'e 3 pour de nombreux types d'hypersurfaces de Fano. Le cas des hypersurfaces cubiques de dimension 4 sur un corps fini reste ouvert.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2019.1.47","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45975748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-16DOI: 10.2140/tunis.2019.1.427
J. Brereton
This paper studies the quintic nonlinear Schr"odinger equation on $mathbb{R}^d$ with randomized initial data below the critical regularity $H^{frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s in (frac{d-2}{2}, frac{d-1}{2})$. The argument further develops the techniques introduced in the work of 'A. B'enyi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.
研究了$mathbb{R}^d$上随机初始数据低于临界正则性$H^{frac{d-1}{2}}$的五次非线性Schr odinger方程。主要结果是给出$H^s$中$s in (frac{d-2}{2}， frac{d-1}{2})$的数据的Wiener随机化，证明了几乎肯定的局部适定性。该论证进一步发展了A的工作中引入的技术。B 'enyi T. Oh和O. Pocovnicu关于三次问题。最后给出了几乎确定全局适定性的一个条件。
{"title":"Almost sure local well-posedness for the supercritical quintic NLS","authors":"J. Brereton","doi":"10.2140/tunis.2019.1.427","DOIUrl":"https://doi.org/10.2140/tunis.2019.1.427","url":null,"abstract":"This paper studies the quintic nonlinear Schr\"odinger equation on $mathbb{R}^d$ with randomized initial data below the critical regularity $H^{frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s in (frac{d-2}{2}, frac{d-1}{2})$. The argument further develops the techniques introduced in the work of 'A. B'enyi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2019.1.427","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-15DOI: 10.2140/tunis.2020.2.399
N. Imai, Yoichi Mieda
In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-supercuspidal parts.
{"title":"Potentially good reduction loci of Shimura varieties","authors":"N. Imai, Yoichi Mieda","doi":"10.2140/tunis.2020.2.399","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.399","url":null,"abstract":"In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-supercuspidal parts.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.399","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68571794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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