Quarterly of Applied Mathematics最新文献

{"title":"A note on the dissipation for the general Muskat problem","authors":"Susanna V. Haziot, B. Pausader","doi":"10.1090/qam/1646","DOIUrl":"https://doi.org/10.1090/qam/1646","url":null,"abstract":"We consider the dissipation of the Muskat problem and we give an elementary proof of a surprising inequality of Constantin-Cordoba-Gancedo-Strain [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 201–227 and Amer. J. Math. 138 (2016), pp. 1455–1494] which holds in greater generality.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48793498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A model of invariant control system using mean curvature drift from Brownian motion under submersions","authors":"Huang Ching-Peng","doi":"10.1090/qam/1633","DOIUrl":"https://doi.org/10.1090/qam/1633","url":null,"abstract":"<p>Given a Riemannian submersion <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"phi colon upper M right-arrow upper N\"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">phi : M to N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct a stochastic process <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the image <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y colon-equal phi left-parenthesis upper X right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>≔</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Y≔phi (X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi colon upper G upper L left-parenthesis n right-parenthesis right-arrow upper G upper L left-parenthesis n right-parenthesis slash upper O left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">pi : GL(n) to GL(n)/O(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose im","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Singularities of the stress concentration in the presence of 𝐶^{1,𝛼}-inclusions with core-shell geometry","authors":"Xia Hao, Zhiwen Zhao","doi":"10.1090/qam/1634","DOIUrl":"https://doi.org/10.1090/qam/1634","url":null,"abstract":"In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance \u0000\u0000 \u0000 ε\u0000 varepsilon\u0000 \u0000\u0000 between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 1\u0000 ,\u0000 α\u0000 \u0000 \u0000 C^{1,alpha }\u0000 \u0000\u0000 boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance \u0000\u0000 \u0000 ε\u0000 varepsilon\u0000 \u0000\u0000 between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44986296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Many-body excitations in trapped Bose gas: A non-Hermitian approach","authors":"M. Grillakis, D. Margetis, S. Sorokanich","doi":"10.1090/qam/1630","DOIUrl":"https://doi.org/10.1090/qam/1630","url":null,"abstract":"We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 \u0000 \u0000 a\u0000 p\u0000 p\u0000 \u0000 \u0000 \u0000 mathcal {H}_{mathrm {app}}\u0000 \u0000\u0000, in the Bosonic Fock space. This \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 \u0000 \u0000 a\u0000 p\u0000 p\u0000 \u0000 \u0000 \u0000 mathcal {H}_{mathrm {app}}\u0000 \u0000\u0000 conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 \u0000 \u0000 a\u0000 p\u0000 p\u0000 \u0000 \u0000 \u0000 mathcal {H}_{mathrm {app}}\u0000 \u0000\u0000. Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a \u0000\u0000 \u0000 J\u0000 J\u0000 \u0000\u0000-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the \u0000\u0000 \u0000 N\u0000 N\u0000 \u0000\u0000-particle sector of Fock space.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44766045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The mathematics of thin structures","authors":"Jean-François Babadjian, Giovanni Di Fratta, I. Fonseca, G. Francfort, M. Lewicka, C. Muratov","doi":"10.1090/qam/1628","DOIUrl":"https://doi.org/10.1090/qam/1628","url":null,"abstract":"This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known models in plate theory (membrane, bending, Von Karmann, etc…), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micromagnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46238051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hydrodynamic alignment with pressure II. Multi-species","authors":"J. Lu, E. Tadmor","doi":"10.1090/qam/1639","DOIUrl":"https://doi.org/10.1090/qam/1639","url":null,"abstract":"We study the long-time hydrodynamic behavior of systems of multi-species which arise from agent-based description of alignment dynamics. The interaction between species is governed by an array of symmetric communication kernels. We prove that the crowd of different species flocks towards the mean velocity if (i) cross interactions form a heavy-tailed connected array of kernels, while (ii) self-interactions are governed by kernels with singular heads. The main new aspect here is that flocking behavior holds without closure assumption on the specific form of pressure tensors. Specifically, we prove the long-time flocking behavior for connected arrays of multi-species, with self-interactions governed by entropic pressure laws (see E. Tadmor [Bull. Amer. Math. Soc. (2023), to appear]) and driven by fractional \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-alignment. In particular, it follows that such multi-species hydrodynamics approaches a mono-kinetic description. This generalizes the mono-kinetic, “pressure-less” study by He and Tadmor [Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), pp. 1031–1053].","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45588342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Riemann problem for a two-phase mixture hyperbolic system with phase function and multi-component equation of state","authors":"M. Hantke, Christoph Matern, G. Warnecke, Hazem Yaghi","doi":"10.1090/qam/1664","DOIUrl":"https://doi.org/10.1090/qam/1664","url":null,"abstract":"In this paper a hyperbolic system of partial differential equations for two-phase mixture flows with \u0000\u0000 \u0000 N\u0000 N\u0000 \u0000\u0000 components is studied. It is derived from a more complicated model involving diffusion and exchange terms. Important features of the model are the assumption of isothermal flow, the use of a phase field function to distinguish the phases and a mixture equation of state involving the phase field function as well as an affine relation between partial densities and partial pressures in the liquid phase. This complicates the analysis. A complete solution of the Riemann initial value problem is given. Some interesting examples are suggested as benchmarks for numerical schemes.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44433071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence, uniqueness, and long-time behavior of linearized field dislocation dynamics","authors":"A. Acharya, M. Slemrod","doi":"10.1090/qam/1642","DOIUrl":"https://doi.org/10.1090/qam/1642","url":null,"abstract":"This paper examines a system of partial differential equations describing dislocation dynamics in a crystalline solid. In particular we consider dynamics linearized about a state of zero stress and use linear semigroup theory to establish existence, uniqueness, and time-asymptotic behavior of the linear system.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43109531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linear stability of liquid Lane-Emden stars","authors":"K. Lam","doi":"10.1090/qam/1677","DOIUrl":"https://doi.org/10.1090/qam/1677","url":null,"abstract":"<p>We establish various qualitative properties of liquid Lane-Emden stars in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, including bounds for its density profile <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho\">\u0000 <mml:semantics>\u0000 <mml:mi>ρ<!-- ρ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">rho</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and radius <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than-or-equal-to 2 left-parenthesis d minus 1 right-parenthesis slash d\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">gamma geq 2(d-1)/d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>; linearly stable when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than 2 left-parenthesis d minus 1 right-parenthesis slash d\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">gamma >2(d-1)/d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for stars with small relative central density <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho left-parenthesis 0 right-parenthesis minus rho left-parenthesis ","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41600692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Oberbeck–Boussinesq system with non-local boundary conditions","authors":"A. Abbatiello, E. Feireisl","doi":"10.1090/qam/1635","DOIUrl":"https://doi.org/10.1090/qam/1635","url":null,"abstract":"<p>We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma upper T Subscript normal m normal a normal x Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">m</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">a</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">x</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[0, T_{mathrm {max}})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Moreover, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Subscript normal m normal a normal x Baseline equals normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">m</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">a</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">x</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">T_{mathrm {max}} = infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the two-dimensional setting.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42223331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}