Following is the list of journals listing in the alphabet order with the shortest peer-review that I have ever seen. If you know more, please drop me a comment below. Please take note that neither survey nor review article is considered in this situation.

- Acta Mathematica

http://www.springerlink.com/content/0001-5962/ - Acta Mathematica Vietnamica

http://link.springer.com/journal/40306

The shortest is**12**days, check this paper by del Angel, Mueller-Stach, van Straten, and Zuo. - Advances in Calculus of Variations

http://www.degruyter.com/view/j/acv

The shortest is**14**days, check this paper by Ejiri and Micallef.

BASED ON RECEIVING AND REVISING DAYS - Advances in Mathematics (AiM)

http://www.sciencedirect.com/science/journal/aip/00018708

The shortest is**3**days, check this paper by Ohta and Sturm.

The runner up is**7**days, check this paper by Li and Nguyen. - American Journal of Mathematics (AJM)

http://muse.jhu.edu/journals/ajm/ - Annales de l’Institut Henri Poincare (C) Non Linear Analysis

http://www.sciencedirect.com/science/journal/aip/02941449

The shortest is**7**days, check this paper by Cagnettia, Gomesa, Mitakec and Tran.

The runner up is**18**days, check this paper by Dolbeault and Toscani and this paper by Ignat, Nguyen, Slastikov, and Zarnescud. - Annales Henri Poincaré

http://link.springer.com/journal/23

The shortest is**0**days, check this paper by Friedrich. - Annali della Scuola Normale Superiore di Pisa (New series)

http://annaliscienze.sns.it/index.php?page=OnlineIndex

The shortest is**80**days, check this paper by Phuc. - Annali di Matematica Pura ed Applicata

http://www.springerlink.com/content/108198/

The shortest is**31**days, check this paper by Batkam and Colin. - Annals of Mathematics (AoM)

http://annals.math.princeton.edu/ - Archive for Rational Mechanics and Analysis (ARMA)

http://www.springerlink.com/content/0003-9527/

The shortest is**43**days, check this paper by Nam and Seiringer. - Bulletin des Sciences Mathématiques

http://www.sciencedirect.com/science/journal/aip/00074497

The shortest is**27**days, check this paper by Cotsiolis and Labropoulos. - Calculus of Variations and Partial Differential Equations

http://www.springerlink.com/content/0944-2669/

The shortest is**10**days, check this paper by Chang, Gursky, and Yang.

The runner-up is**28**days, check this paper by Ros-Oton and Serra. - Communications in Contemporary Mathematics (CCM)

http://www.worldscientific.com/worldscinet/ccm

The shortest is**19**days, check this paper by M.M. Fall and T. Weth.

The runner-up is**27**days, check this paper by S.Y. Yolcu and T. Yolcu.

The third place is**33**days, check this paper by Rong and Zhang. - Communications in Mathematical Physics (CMP)

http://www.springerlink.com/content/100467/

The shortest is**20**days, check this paper by Constantin, Tarfulea, and Vicol .

The runner-up is**37**days, check this paper by Dybalski and Gérard.

The third place is**43**days, check this paper by Galloway and Schoen and this paper by Hebey, Pacard, and Pollack. - Communications in Partial Differential Equations

http://www.tandfonline.com/toc/lpde20/current - Communications on Pure and Applied Mathematics (CPAM)

http://onlinelibrary.wiley.com/journal/10.1002/%28ISSN%291097-0312

NO ACCEPTANCE DATE SHOWN - Comptes Rendus Mathematique

http://www.sciencedirect.com/science/journal/aip/1631073X

The shortest is**0**day, check this paper by Druet.

The runner-up is**1**day, check this paper by Mawhina, Ortegab, and Robles-Pérezb. - Duke Mathematical Journal (DMJ)

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&page=past&handle=euclid.dmj

The shortest is**11**days, check this paper by Franecki and Kapranov.

The runner-up is**15**days, check this paper by Šilhavý.

The third place is**28**days, check this paper by Lin and Rivière.

The fourth place is about**31**days, check this paper by Farb, Lubotzky, and Minsky.

NO ACCEPTANCE DATE SHOWN, ONLY REVISION DATE SHOWN - Geometric And Functional Analysis (GAFA)

http://www.springerlink.com/content/1016-443x/

The shortest is**28**days, check this paper by Alon and Bourgain.

The runner-up is**42**days, check this paper by Bourgain, Dirksen, and Nelson.

The third place is**45**days, check this paper by Bourgain and Guth.

The fourth place is about**50**days, check this paper by Tao and Vu.

(Ref.: a comment below.) - Indiana University Mathematics Journal

http://www.iumj.indiana.edu/IUMJ/issues.php - International Journal of Mathematics

http://www.worldscientific.com/worldscinet/ijm

The shortest is**1**days, check this paper by Fang, Jiang, Lin, and Xu. - International Mathematics Research Notices

http://imrn.oxfordjournals.org/content/by/year

The shortest is**2**days, check this paper by Hang and Yang.

The runner-up is**54**days, check this paper by Bourgain. - Inventiones Mathematicae

http://www.springerlink.com/content/0020-9910/

The shortest is**31**days, check this paper by Venerucci. - Journal de Mathématiques Pures et Appliquées (JMPA)

http://www.sciencedirect.com/science/journal/aip/00217824

NO ACCEPTANCE DATE SHOWN, ONLY AVAILABILITY DATE SHOWN - Journal für die reine und angewandte Mathematik (Crelle)

http://www.degruyter.com/view/j/crll?aheadOfPrint

The shortest is**66**days, check this paper by Tian and Zhu. - Journal of Differential Equations (JDE)

http://www.sciencedirect.com/science/journal/aip/00220396

The shortest is**43**days, check this paper by Byun, Ok, and Ryu. - Journal of Differential Geometry (JDG)

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&page=past&handle=euclid.jdg

NO ACCEPTANCE DATE SHOWN - Journal of Functional Analysis (JFA)

http://www.sciencedirect.com/science/journal/aip/00221236

The shortest is**0**days, check this paper by Mendelson and Paouris.

The runner-up is**2**days, check this paper by Bourgain and Bulut.

The third place is**7**days, check this paper by Bidaut-Véron, Chasseigne, and Véron.

The fourth place is**11**days, check this paper by Escobar and Garcia.

The fifth place is**13**days, check this paper by Aubin. - Journal of Geometric Analysis

http://www.springerlink.com/content/1050-6926/ - Journal of Global Optimization

http://link.springer.com/journal/10898

The shortest is**0**days, check this paper by Colombo and Wolenski. - Journal of Mathematical Analysis and Applications

http://www.sciencedirect.com/science/journal/aip/0022247X - Journal of Number Theory

http://www.sciencedirect.com/science/journal/0022314X

The shortest is**7**days, check this paper by Vinh. - Journal of the American Mathematical Society (JAMS)

http://www.ams.org/publications/journals/journalsframework/jams - Journal of the European Mathematical Society (JEMS)

http://www.ems-ph.org/journals/all_issues.php?issn=1435-9855

NO ACCEPTANCE DATE SHOWN, ONLY REVISION DATE SHOWN - manuscripta mathematica

http://www.springerlink.com/content/0025-2611/ - Mathematical Research Letters

http://intlpress.com/site/pub/pages/journals/items/mrl/_home/_main/index.html

NO ACCEPTANCE DATE SHOWN

- Mathematische Annalen

http://www.springerlink.com/content/0025-5831/ - Mathematische Zeitschrift

http://www.springerlink.com/content/0025-5874/ - Memoirs of the American Mathematical Society

http://www.ams.org/publications/journals/journalsframework/memo - NoDEA: Nonlinear Differential Equations and Applications

http://www.springerlink.com/content/1021-9722/ - Nonlinear Analysis: Theory, Methods & Applications

http://www.sciencedirect.com/science/journal/aip/0362546X

The shortest is**0**day, check this paper by Garcia-Azoreroa, Manfredib, Perala, and Rossi.

The runner-up is**9**days, check this paper by Chiron; but I guess there should have a paper having less than 9 days. - Proceedings of the American Mathematical Society

http://www.ams.org/jourcgi/jrnl_toolbar_nav/proc_all - Publications Mathématiques de L’IHÉS

http://www.springerlink.com/content/0073-8301/ - Set-Valued and Variational Analysis

http://link.springer.com/journal/11228

The shortest is**1**day, check this paper by Bonnans and de la Vega - SIAM Journal on Mathematical Analysis

http://epubs.siam.org/toc/sjmaah/current - The Taiwanese Journal of Mathematics

http://journal.taiwanmathsoc.org.tw

The shortest is**6**days, check this paper by Chen. - Transactions of the American Mathematical Society

http://www.ams.org/jourcgi/jrnl_toolbar_nav/tran_all - Vietnam Journal of Mathematics

http://link.springer.com/journal/10013

The shortest is**11**days, check this paper by Phu.

The runner-up is**49**days, check this paper by Loi.

I take this chance to acknowledge the contribution from

- Dr. Nguyen Thac Dung from the National Taiwan University;
- Mr. Nguyen Van Luong from the University of Padova;
- Mr. Duong Duc Lam from the University of Sussex;
- …

for their contribution to the list/record.

more information on JFA: http://www.sciencedirect.com/science/article/pii/S002212361200050X#

Comment by Hoang Van Ngoc — November 4, 2013 @ 21:42

Thanks Ngoc, it’s great. I will update right now.

Comment by Ngô Quốc Anh — November 4, 2013 @ 22:01

For CMP, there are some more information

http://link.springer.com/article/10.1007/s00220-016-2593-z (20 days)

http://link.springer.com/article/10.1007%2Fs00220-012-1631-8 (35 days)

Comment by Duong Duc Lam — March 11, 2016 @ 1:10

Great!

Comment by Ngô Quốc Anh — March 11, 2016 @ 1:36

Dear QA,

I think you gave a wrong link for NA, it should be: http://www.sciencedirect.com/science/article/pii/S0362546X05009867

Another candidate for short papers (of NA) is: http://www.sciencedirect.com/science/article/pii/S0362546X04001671 (by Chiron, 9 days)

For GAFA, the following might be in the shortest list:

http://link.springer.com/article/10.1007/s00039-014-0270-y (by Alon and JB, 28 days)

http://link.springer.com/article/10.1007/s00039-015-0332-9 (by JB, Dirksen and Nelson, 42 days)

http://link.springer.com/article/10.1007/s00039-011-0140-9 (by JB and Guth, 45 days)

http://link.springer.com/article/10.1007/s00039-010-0057-8 (by Tao and Vu, 50 days)

Comment by Duong Duc Lam — July 17, 2016 @ 17:11

Thanks Lam, I will make correction ASAP🙂. Please keep informing me interesting news.

Comment by Ngô Quốc Anh — July 17, 2016 @ 19:02

Another runner-up for Annales de l’Institut Henri Poincare (C) Non Linear Analysis

http://www.sciencedirect.com/science/article/pii/S0294144915000402

Comment by Duong Duc Lam — July 19, 2016 @ 18:48

Update for Nonlinear Analysis: This one might be the runner-up, Dr Ngo

(4 days by Brezis, Nguyen) http://www.sciencedirect.com/science/article/pii/S0362546X16000407

Comment by Duong Duc Lam — September 5, 2016 @ 0:20