In this paper we present the analytical study of the magnetohydrodynamic (MHD) flow through a rectangular duct driven by the pressure gradient and under the action of the transverse magnetic field. Motivated by various MHD applications in which hydrodynamic slip naturally occur, we prescribe the slipping boundary condition on the upper boundary which contains irregularities as well. Depending on the period of the boundary roughness, we derive three different limit problems by using rigorous analysis in the appropriate functional setting. This approach also enables us to determine the relative contribution of the MHD effect and the slip itself in the governing coupled system satisfied by the velocity and induced magnetic field.
{"title":"The effects of boundary roughness on the MHD duct flow with slip hydrodynamic condition","authors":"Igor Pažanin, Marcone Pereira","doi":"10.1090/qam/1686","DOIUrl":"https://doi.org/10.1090/qam/1686","url":null,"abstract":"In this paper we present the analytical study of the magnetohydrodynamic (MHD) flow through a rectangular duct driven by the pressure gradient and under the action of the transverse magnetic field. Motivated by various MHD applications in which hydrodynamic slip naturally occur, we prescribe the slipping boundary condition on the upper boundary which contains irregularities as well. Depending on the period of the boundary roughness, we derive three different limit problems by using rigorous analysis in the appropriate functional setting. This approach also enables us to determine the relative contribution of the MHD effect and the slip itself in the governing coupled system satisfied by the velocity and induced magnetic field.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140471287","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":"Preface for the first special issue in honor of Bob Pego","authors":"Govind Menon","doi":"10.1090/qam/1673","DOIUrl":"https://doi.org/10.1090/qam/1673","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139229261","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}
In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter δdelta that can decrease from δ=1delta =1 for the Fermi-Dirac particles to δ=0delta =0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space LT∞Lv,x∞∩LT∞Lx∞Lv1L^{infty }_{T}L^{infty }_{v,x}cap L^{infty }_{T}L^{infty }_{x}L^1_v
{"title":"Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials","authors":"Zongguang Li","doi":"10.1090/qam/1681","DOIUrl":"https://doi.org/10.1090/qam/1681","url":null,"abstract":"In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta\"> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that can decrease from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta equals 1\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">delta =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the Fermi-Dirac particles to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">delta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript upper T Superscript normal infinity Baseline upper L Subscript v comma x Superscript normal infinity intersection upper L Subscript upper T Superscript normal infinity Baseline upper L Subscript x Superscript normal infinity Baseline upper L Subscript v Superscript 1\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^{infty }_{T}L^{infty }_{v,x}cap L^{infty }_{T}L^{infty }_{x}L^1_v</mml:annotation> </mml:semantics> </mml:math> </inline-formula","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262530","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}
We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by P=σ2ρP= sigma ^2 rho, where σsigma, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.
{"title":"Self-similar solutions of the relativistic Euler system with spherical symmetry","authors":"Bing-Ze Lu, Chou Kao, Wen-Ching Lien","doi":"10.1090/qam/1680","DOIUrl":"https://doi.org/10.1090/qam/1680","url":null,"abstract":"We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P equals sigma squared rho\"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">P= sigma ^2 rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273963","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}
Shock wave stability for isentropic irrotational flow is studied for Euler system but with shock front conditions corresponding to the second order nonlinear wave equation. It is shown that the usual Lax’ shock condition still guarantees the uniform linear stability and therefore the existence of the shock waves solution.
{"title":"Shock waves with irrotational Rankine-Hugoniot conditions","authors":"Dening Li, Qingtian Zhang","doi":"10.1090/qam/1682","DOIUrl":"https://doi.org/10.1090/qam/1682","url":null,"abstract":"Shock wave stability for isentropic irrotational flow is studied for Euler system but with shock front conditions corresponding to the second order nonlinear wave equation. It is shown that the usual Lax’ shock condition still guarantees the uniform linear stability and therefore the existence of the shock waves solution.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413347","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}
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.
{"title":"Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach","authors":"Uditnarayan Kouskiya, Amit Acharya","doi":"10.1090/qam/1679","DOIUrl":"https://doi.org/10.1090/qam/1679","url":null,"abstract":"A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135855642","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}
We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.
{"title":"Remarks on the linear wave equation","authors":"John Ball","doi":"10.1090/qam/1678","DOIUrl":"https://doi.org/10.1090/qam/1678","url":null,"abstract":"We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973404","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}
The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of 2 2 -line solitons whose line solitons interact elastically each other. Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.
{"title":"Linear stability of elastic 2-line solitons for the KP-II equation","authors":"Tetsu Mizumachi","doi":"10.1090/qam/1676","DOIUrl":"https://doi.org/10.1090/qam/1676","url":null,"abstract":"The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000-line solitons whose line solitons interact elastically each other. Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49028911","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}
We study a two-dimensional quaternary inhibitory system. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which three species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a three-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. Geometrical descriptions of limit configurations are derived.
{"title":"On a quaternary nonlocal isoperimetric problem","authors":"S. Alama, L. Bronsard, Xinyang Lu, Chong Wang","doi":"10.1090/qam/1675","DOIUrl":"https://doi.org/10.1090/qam/1675","url":null,"abstract":"We study a two-dimensional quaternary inhibitory system. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which three species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a three-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. Geometrical descriptions of limit configurations are derived.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42884797","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}
This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called σ2,psigma _{2,p}-energy, in polar coordinates. Let X⊂Rn{mathbb {X}}subset mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional (1.1)(1.1) whose integrand is defined by W(∇u(x))≔(σ2
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