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Journal articles on the topic 'Gagliardo-Nirenberg inequality'

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Published: 4 June 2021
Last updated: 3 February 2022

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1

Huang, Qingzhong, and Ai-Jun Li. "The L Gagliardo-Nirenberg-Zhang inequality." Advances in Applied Mathematics 113 (February 2020): 101971. http://dx.doi.org/10.1016/j.aam.2019.101971.

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2

Ceccon, Jurandir, and Marcos Montenegro. "General optimal euclidean Sobolev and Gagliardo-Nirenberg inequalities." Anais da Academia Brasileira de Ciências 77, no. 4 (2005): 581–87. http://dx.doi.org/10.1590/s0001-37652005000400001.

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We prove general optimal euclidean Sobolev and Gagliardo-Nirenberg inequalities by using mass transportation and convex analysis results. Explicit extremals and the computation of some optimal constants are also provided. In particular we extend the optimal Gagliardo-Nirenberg inequality proved by Del Pino and Dolbeault 2003 and the optimal inequalities proved by Cordero-Erausquin et al. 2004.
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3

Esfahani, Amin. "Anisotropic Gagliardo–Nirenberg inequality with fractional derivatives." Zeitschrift für angewandte Mathematik und Physik 66, no. 6 (2015): 3345–56. http://dx.doi.org/10.1007/s00033-015-0586-y.

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4

Lokharu, E. "Gagliardo–Nirenberg inequality for maximal functions measuring smoothness." Journal of Mathematical Sciences 182, no. 5 (2012): 663–73. http://dx.doi.org/10.1007/s10958-012-0771-x.

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5

Dolbeault, Jean, and Maria J. Esteban. "Extremal functions for Caffarelli—Kohn—Nirenberg and logarithmic Hardy inequalities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 4 (2012): 745–67. http://dx.doi.org/10.1017/s0308210510001101.

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We consider a family of Caffarelli–Kohn–Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities that were obtained recently as a limit case of the Caffarelli–Kohn–Nirenberg inequalities. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo–Nirenberg interpolation inequalities and Gross's logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.
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6

Kögler, Kevin, and Phan Thành Nam. "The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit." Archive for Rational Mechanics and Analysis 240, no. 3 (2021): 1169–202. http://dx.doi.org/10.1007/s00205-021-01633-8.

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AbstractWe consider an analogue of the Lieb–Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb–Thirring constant converges to the optimal constant of the one-body Gagliardo–Nirenberg interpolation inequality without interaction.
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7

Wu, Xiulan, and Jun Fu. "Extinction and Decay Estimates of Solutions for thep-Laplacian Equations with Nonlinear Absorptions and Nonlocal Sources." Abstract and Applied Analysis 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/928080.

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We investigate the extinction and decay estimates of thep-Laplacian equations with nonlinear absorptions and nonlocal sources. By Gagliardo-Nirenberg inequality, we obtain the sufficient conditions of extinction solutions, and we also give the precise decay estimates of the extinction solutions.
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8

Kovalevsky, Alexander, and Francesco Nicolosi. "A weighted interpolation inequality of the Nirenberg–Gagliardo kind." Nonlinear Analysis: Theory, Methods & Applications 36, no. 3 (1999): 269–73. http://dx.doi.org/10.1016/s0362-546x(97)00625-1.

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9

Fiorenza, Alberto, Maria Rosaria Formica, Tomáš Roskovec, and Filip Soudský. "Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces." Rendiconti Lincei - Matematica e Applicazioni 30, no. 4 (2019): 847–64. http://dx.doi.org/10.4171/rlm/872.

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10

Nguyen, Van Hoang. "The sharp Gagliardo–Nirenberg–Sobolev inequality in quantitative form." Journal of Functional Analysis 277, no. 7 (2019): 2179–208. http://dx.doi.org/10.1016/j.jfa.2019.02.016.

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11

Kopaliani, Tengiz, and George Chelidze. "Gagliardo–Nirenberg type inequality for variable exponent Lebesgue spaces." Journal of Mathematical Analysis and Applications 356, no. 1 (2009): 232–36. http://dx.doi.org/10.1016/j.jmaa.2009.03.012.

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12

Li, Jiaojiao, and Li Ma. "Extremals to new Gagliardo–Nirenberg inequality and ground states." Applied Mathematics Letters 120 (October 2021): 107266. http://dx.doi.org/10.1016/j.aml.2021.107266.

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13

Ruzhansky, Michael, Durvudkhan Suragan, and Nurgissa Yessirkegenov. "Euler semigroup, Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups." Semigroup Forum 101, no. 1 (2020): 162–91. http://dx.doi.org/10.1007/s00233-020-10110-9.

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Abstract In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ { e - t E ∗ E } t > 0 on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator $$\mathbb {E}$$ E . Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $$|\cdot |$$ | · | -radial weighted Hardy–Sobolev type inequality are established.
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14

Lin, Jeng-Eng. "Nonexistence of Nontrivial Stationary Solutions with Decay Order for Some Nonlinear Evolution Equations." Journal of Mathematics Research 9, no. 3 (2017): 1. http://dx.doi.org/10.5539/jmr.v9n3p1.

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We show that there are no nontrivial stationary solutions of certain decay order for some applied nonlinear evolution equations which include the thin epitaxial film model with slope selection and the square phase field crystal (SPFC) equation. The method is to use the Morawetz multiplier and the Gagliardo-Nirenberg inequality.
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15

Brouttelande, Christophe. "THE BEST-CONSTANT PROBLEM FOR A FAMILY OF GAGLIARDO–NIRENBERG INEQUALITIES ON A COMPACT RIEMANNIAN MANIFOLD." Proceedings of the Edinburgh Mathematical Society 46, no. 1 (2003): 117–46. http://dx.doi.org/10.1017/s0013091501000426.

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AbstractThe best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.AMS 2000 Mathematics subject classification: Primary 58J05
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16

Huh, Hyungjin, and Bora Moon. "Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field." Advances in Mathematical Physics 2018 (June 4, 2018): 1–8. http://dx.doi.org/10.1155/2018/3962062.

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We study the Cauchy problem of the Chern-Simons-Schrödinger equations with a neutral field, under the Coulomb gauge condition, in energy space H1(R2). We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.
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17

Kałamajska, Agnieszka, and Katarzyna Pietruska-Pałuba. "On a Variant of the Gagliardo–Nirenberg Inequality Deduced from the Hardy Inequality." Bulletin of the Polish Academy of Sciences Mathematics 59, no. 2 (2011): 133–49. http://dx.doi.org/10.4064/ba59-2-4.

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18

Bolley, François, Dario Cordero-Erausquin, Yasuhiro Fujita, Ivan Gentil, and Arnaud Guillin. "New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality." International Mathematics Research Notices 2020, no. 10 (2018): 3042–83. http://dx.doi.org/10.1093/imrn/rny111.

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Abstract We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge between the geometric point of view of the Brunn–Minkowski inequality and the functional point of view of the Sobolev-type inequalities. In this way we unify, simplify, and generalize results by S. Bobkov–M. Ledoux, M. del Pino–J. Dolbeault, and B. Nazaret.
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19

Moonens, Laurent, and Tiago H. Picon. "Solving the Equation." Proceedings of the Edinburgh Mathematical Society 61, no. 4 (2018): 1055–61. http://dx.doi.org/10.1017/s0013091518000172.

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AbstractIn the following note, we focus on the problem of existence of continuous solutions vanishing at infinity to the equation div v = f for f ∈ Ln(ℝn) and satisfying an estimate of the type ||v||∞ ⩽ C||f||n for any f ∈ Ln(ℝn), where C > 0 is related to the constant appearing in the Sobolev–Gagliardo–Nirenberg inequality for functions with bounded variation (BV functions).
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20

Caspers, Wim, and Guido Sweers. "Point interactions on bounded domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 5 (1994): 917–26. http://dx.doi.org/10.1017/s0308210500022411.

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The Laplacian operator Δ on a bounded domain Ω in ℝn containing 0, with Dirichlet boundary condition, is perturbed by a pseudopotential δ, the Dirac measure at 0. Such a perturbation will be defined in Lp(ℝ) for n = 2, 1 <lt; p < ∞, and for n = 3, < p < 3, and is shown to be the generator of an analytic semigroup. Thus solutions of the corresponding evolutionary system are well defined. The necessary estimates involve the Gagliardo– Nirenberg inequality and the Kato inequality.
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21

Capone, Claudia, Alberto Fiorenza, and Agnieszka Kałamajska. "Strongly nonlinear Gagliardo–Nirenberg inequality in Orlicz spaces and Boyd indices." Rendiconti Lincei - Matematica e Applicazioni 28, no. 1 (2017): 119–41. http://dx.doi.org/10.4171/rlm/755.

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22

Miyazaki, Yoichi. "A short proof of the Gagliardo-Nirenberg inequality with BMO term." Proceedings of the American Mathematical Society 148, no. 10 (2020): 4257–61. http://dx.doi.org/10.1090/proc/15048.

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23

Chen, Jianqing, and Boling Guo. "Sharp constant of an improved Gagliardo–Nirenberg inequality and its application." Annali di Matematica Pura ed Applicata 190, no. 2 (2010): 341–54. http://dx.doi.org/10.1007/s10231-010-0152-3.

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24

Esfahani, Amin, and Ademir Pastor. "Sharp Constant of an Anisotropic Gagliardo–Nirenberg-Type Inequality and Applications." Bulletin of the Brazilian Mathematical Society, New Series 48, no. 1 (2016): 171–85. http://dx.doi.org/10.1007/s00574-016-0017-5.

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25

Fiorenza, Alberto, Maria Rosaria Formica, Tomáš Roskovec, and Filip Soudský. "Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks." Zeitschrift für Analysis und ihre Anwendungen 40, no. 2 (2021): 217–36. http://dx.doi.org/10.4171/zaa/1681.

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26

HAMZA, MOHAMED ALI, and HATEM ZAAG. "LYAPUNOV FUNCTIONAL AND BLOW-UP RESULTS FOR A CLASS OF PERTURBATIONS OF SEMILINEAR WAVE EQUATIONS IN THE CRITICAL CASE." Journal of Hyperbolic Differential Equations 09, no. 02 (2012): 195–221. http://dx.doi.org/10.1142/s0219891612500063.

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We study a class of perturbations for the semilinear wave equation with critical power nonlinearity (in the conformal transform sense). Working in the framework of similarity variables, we introduce a Lyapunov functional for this problem. Using a two-step argument based on interpolation and a critical Gagliardo–Nirenberg inequality, we establish that the blow-up rate of any singular solution is given by the solution of the nonperturbed associated ODE, specifically u″ = up.
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27

Cianci, Paolo, and Francesco Nicolosi. "A weighted interpolation inequality with variable exponent of the Nirenberg–Gagliardo kind." Nonlinear Analysis: Theory, Methods & Applications 71, no. 12 (2009): 5915–29. http://dx.doi.org/10.1016/j.na.2009.05.015.

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28

Sawano, Yoshihiro, and Hidemitsu Wadade. "On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space." Journal of Fourier Analysis and Applications 19, no. 1 (2012): 20–47. http://dx.doi.org/10.1007/s00041-012-9223-8.

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29

Kozono, Hideo, and Hidemitsu Wadade. "Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO." Mathematische Zeitschrift 259, no. 4 (2007): 935–50. http://dx.doi.org/10.1007/s00209-007-0258-5.

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30

Mizuta, Yoshihiro, Eiichi Nakai, Yoshihiro Sawano, and Tetsu Shimomura. "Gagliardo–Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces." Archiv der Mathematik 98, no. 3 (2012): 253–63. http://dx.doi.org/10.1007/s00013-012-0362-6.

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31

Nicolosi, F., and Igor V. Skrypnik. "Nirenberg-Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations." Topological Methods in Nonlinear Analysis 7, no. 2 (1996): 327. http://dx.doi.org/10.12775/tmna.1996.015.

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32

Porretta, Alessio. "A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁". Advanced Nonlinear Studies 20, № 2 (2020): 361–71. http://dx.doi.org/10.1515/ans-2020-2086.

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AbstractIt is known that the Sobolev space {W^{1,p}(\mathbb{R}^{N})} is embedded into {L^{Np/(N-p)}(\mathbb{R}^{N})} if {p<N} and into {L^{\infty}(\mathbb{R}^{N})} if {p>N}. There is usually a discontinuity in the proof of those two different embeddings since, for {p>N}, the estimate {\lVert u\rVert_{\infty}\leq C\lVert Du\rVert_{p}^{N/p}\lVert u\rVert_{p}^{1-N% /p}} is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the {L^{\infty}}-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate {\lVert u\rVert_{N/(N-1)}\leq C\lVert Du\rVert_{1}}. This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.
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33

Bartuccelli, Michele, Jonathan Deane, and Sergey Zelik. "Asymptotic expansions and extremals for the critical Sobolev and Gagliardo–Nirenberg inequalities on a torus." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 143, no. 3 (2013): 445–82. http://dx.doi.org/10.1017/s0308210511000473.

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We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.
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34

Du, Miao, Lixin Tian, Jun Wang, and Fubao Zhang. "Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 03 (2018): 617–53. http://dx.doi.org/10.1017/prm.2018.41.

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AbstractIn this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.
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35

MIZUTA, Yoshihiro, Eiichi NAKAI, Yoshihiro SAWANO, and Tetsu SHIMOMURA. "Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials." Journal of the Mathematical Society of Japan 65, no. 2 (2013): 633–70. http://dx.doi.org/10.2969/jmsj/06520633.

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36

Maz'ya, Vladimir, and Tatyana Shaposhnikova. "On the Brezis and Mironescu conjecture concerning a Gagliardo–Nirenberg inequality for fractional Sobolev norms." Journal de Mathématiques Pures et Appliquées 81, no. 9 (2002): 877–84. http://dx.doi.org/10.1016/s0021-7824(02)01262-x.

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37

Kałamajska, Agnieszka, and Jan Peszek. "On some nonlinear extensions of the Gagliardo–Nirenberg inequality with applications to nonlinear eigenvalue problems." Asymptotic Analysis 77, no. 3-4 (2012): 169–96. http://dx.doi.org/10.3233/asy-2011-1079.

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38

Kozono, Hideo, Tokushi Sato, and Hidemitsu Wadade. "Upper bound of the best constant of the Trudinger-Moser inequality and its application to the Gagliardo-Nirenberg inequality." Indiana University Mathematics Journal 55, no. 6 (2006): 1951–74. http://dx.doi.org/10.1512/iumj.2006.55.2743.

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39

Abbrescia, Leonardo Enrique, and Willie Wai Yeung Wong. "Global versions of the Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations." Transactions of the American Mathematical Society 374, no. 2 (2020): 773–802. http://dx.doi.org/10.1090/tran/8277.

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40

Kałamajska, Agnieszka, and Jan Peszek. "On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities." Journal of Fixed Point Theory and Applications 13, no. 1 (2013): 271–90. http://dx.doi.org/10.1007/s11784-013-0106-7.

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41

Georgiev, Vladimir, and Sandra Lucente. "Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential." Journal of Hyperbolic Differential Equations 15, no. 04 (2018): 755–88. http://dx.doi.org/10.1142/s0219891618500248.

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We study the dynamics for the focusing nonlinear Klein–Gordon equation, [Formula: see text] with positive radial potential [Formula: see text] and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on [Formula: see text]. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.
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42

Hashimoto, Daiki, Yoshihiro Sawano, and Tetsu Shimomura. "Gagliardo--Nirenberg inequality for generalized Riesz potentials of functions in Musielak--Orlicz spaces over quasi-metric measure spaces." Colloquium Mathematicum 161, no. 1 (2020): 51–66. http://dx.doi.org/10.4064/cm7535-4-2019.

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43

Choczewski, Tomasz, and Agnieszka Kalamajska. "On one variant of strongly nonlinear Gagliardo–Nirenberg inequality involving Laplace operator with application to nonlinear elliptic problems." Rendiconti Lincei - Matematica e Applicazioni 30, no. 3 (2019): 479–96. http://dx.doi.org/10.4171/rlm/856.

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44

Fila, Marek, and Michael Winkler. "A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation." Advances in Mathematics 357 (December 2019): 106823. http://dx.doi.org/10.1016/j.aim.2019.106823.

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45

Khan, Sudheer, Wang Shu, and Monica Abhidha. "Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient." Science Journal of Applied Mathematics and Statistics 8, no. 5 (2020): 53. http://dx.doi.org/10.11648/j.sjams.20200805.11.

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46

Wadade, Hidemitsu. "Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case." Journal of Fourier Analysis and Applications 15, no. 6 (2009): 857–70. http://dx.doi.org/10.1007/s00041-009-9069-x.

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47

Frank, Rupert L., David Gontier, and Mathieu Lewin. "The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities." Communications in Mathematical Physics 384, no. 3 (2021): 1783–828. http://dx.doi.org/10.1007/s00220-021-04039-5.

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AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.
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48

Wu, Jie. "Global Existence of the Three Dimensional Heat-conductive Incompressible Viscous Fluids." MATEC Web of Conferences 179 (2018): 01001. http://dx.doi.org/10.1051/matecconf/201817901001.

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In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in ℝ3. About the heat-conducting incompressible viscous fluids, there are many mathematical researchers study the variants systems when the viscosity and heat-conductivity coefficient are positive. For the heat-conductive system, it is difficulty to get the better regularity due to the gradient of velocity of fluid own the higher order term. It is hard to control it. In order to get its global solutions, we must obtain the a priori estimates at first, then using fixed point theorem, it need the mapping is contracted. We can get a local solution, then applying the criteria extension. We can extend the local solution to the global solutions. For the two dimensional case, the Gagliardo-Nirenberg interpolation inequality makes use of better than the three dimensional situation. Thus, our problem will become more difficulty to handle. In this paper, we assume the coefficient of viscosity is a constant and the coefficient of heat-conductivity satisfying some suitable conditions. We show that the Cauchy problem has a global-in-time strong solution (u,θ) on ℝ3 ×(0, ∞).
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49

Nguyen, Van Hoang. "Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)". Communications in Contemporary Mathematics 21, № 04 (2019): 1850023. http://dx.doi.org/10.1142/s0219199718500232.

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We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].
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Li, Tong, Anthony Suen, Michael Winkler, and Chuan Xue. "Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms." Mathematical Models and Methods in Applied Sciences 25, no. 04 (2015): 721–46. http://dx.doi.org/10.1142/s0218202515500177.

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Abstract:
We study non-negative solutions to the chemotaxis system [Formula: see text] under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞)2 with values in [0, ∞) and ℝ2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (⋆), in the special case [Formula: see text] reducing to a version of the standard Keller–Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by [Formula: see text], reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u0, v0) fulfilling a smallness condition on the norm of v0 in L∞(Ω), the corresponding initial-boundary value problem associated with (⋆) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo–Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) ≡ (μ, κ) in the large time limit, with μ := fΩu0 and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.
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