The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.
{"title":"The anisotropic Cahn–Hilliard equation: Regularity theory and strict separation properties","authors":"H. Garcke, P. Knopf, J. Wittmann","doi":"10.3934/dcdss.2023146","DOIUrl":"https://doi.org/10.3934/dcdss.2023146","url":null,"abstract":"The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"31 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89481144","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}
Boštjan Gabrovšek, Giovanni Molica Bisci, Dušan D. Repovš
In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term begin{document}$ f $end{document} has a suitable oscillating behaviour either at the origin or at infinity.
In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term begin{document}$ f $end{document} has a suitable oscillating behaviour either at the origin or at infinity.
{"title":"On nonlocal Dirichlet problems with oscillating term","authors":"Boštjan Gabrovšek, Giovanni Molica Bisci, Dušan D. Repovš","doi":"10.3934/dcdss.2022130","DOIUrl":"https://doi.org/10.3934/dcdss.2022130","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term <inline-formula><tex-math id=\"M1\">begin{document}$ f $end{document}</tex-math></inline-formula> has a suitable oscillating behaviour either at the origin or at infinity.</p>","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"121 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78428342","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 a phase-field model which describes the interactions between the blood flow and the thrombus. The latter is supposed to be a viscoelastic material. The potential describing the cohesive energy of the mixture is assumed to be of Flory-Huggins type (i.e. logarithmic). This ensures the boundedness from below of the dissipation energy. In the two dimensional case, we prove the local (in time) existence and uniqueness of a strong solution, provided that the two viscosities of the pure fluid phases are close enough. We also show that the order parameter remains strictly separated from the pure phases if it is so at the initial time.
{"title":"A phase-field system arising from multiscale modeling of thrombus biomechanics in blood vessels: Local well-posedness in dimension two","authors":"M. Grasselli, A. Poiatti","doi":"10.3934/dcdss.2023105","DOIUrl":"https://doi.org/10.3934/dcdss.2023105","url":null,"abstract":"We consider a phase-field model which describes the interactions between the blood flow and the thrombus. The latter is supposed to be a viscoelastic material. The potential describing the cohesive energy of the mixture is assumed to be of Flory-Huggins type (i.e. logarithmic). This ensures the boundedness from below of the dissipation energy. In the two dimensional case, we prove the local (in time) existence and uniqueness of a strong solution, provided that the two viscosities of the pure fluid phases are close enough. We also show that the order parameter remains strictly separated from the pure phases if it is so at the initial time.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"33 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91284359","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}
Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.
{"title":"Degenerate diffusion with Preisach hysteresis","authors":"Chiara Gavioli, Pavel Krejvc'i","doi":"10.3934/dcdss.2023154","DOIUrl":"https://doi.org/10.3934/dcdss.2023154","url":null,"abstract":"Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82380023","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 first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $mathcal{R}_0$.
{"title":"The linear stability and basic reproduction numbers for autonomous FDEs","authors":"Xiao-Qiang Zhao","doi":"10.3934/dcdss.2023082","DOIUrl":"https://doi.org/10.3934/dcdss.2023082","url":null,"abstract":"In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $mathcal{R}_0$.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"10 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063197","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}
Martin L. Kliemank, D. Wilde, Mario Bedrunka, A. Krämer, H. Foysi, D. Reith
{"title":"Assessment of lattice Boltzmann method for low-rise building wind flow simulation with limited resources","authors":"Martin L. Kliemank, D. Wilde, Mario Bedrunka, A. Krämer, H. Foysi, D. Reith","doi":"10.3934/dcdss.2023046","DOIUrl":"https://doi.org/10.3934/dcdss.2023046","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"171 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77495628","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":"Uniqueness of stationary distribution and exponential convergence for distribution dependent SDEs","authors":"Shao-Qin Zhang","doi":"10.3934/dcdss.2023003","DOIUrl":"https://doi.org/10.3934/dcdss.2023003","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"5 9 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76526629","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":"Concentration phenomenon for a fractional Schrödinger equation with discontinuous nonlinearity","authors":"V. Ambrosio","doi":"10.3934/dcdss.2023074","DOIUrl":"https://doi.org/10.3934/dcdss.2023074","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"151 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76856438","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":"Nonradial solutions for coupled elliptic system with critical exponent in exterior domain","authors":"Yuxia Guo, Dewei Li","doi":"10.3934/dcdss.2023099","DOIUrl":"https://doi.org/10.3934/dcdss.2023099","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"26 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86968739","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 two-relaxation-time collision operator in discrete kinetic theory models collisions between particles by grouping them into pairs with anti-parallel velocities. It prescribes a linear relaxation towards equilibrium with one rate for the even combination of distribution functions for each pair, and another rate for the odd combination. We reformulate this collision operator using relaxation rates for the forward-propagating and backward-propagating combinations instead. An optimal pair of relaxation rates sets the forward-propagating combination of each pair of distributions to equilibrium. Only the backward-propagating non-equilibrium distributions remain. Applying this result twice gives closed discrete equations for evolving the macroscopic variables alone across three time levels. We split the equivalent equations into a first-order system: a conservation law and a kinetic equation for the flux. All other quantities are evaluated at equilibrium. We apply this formalism to the magnetic field in a lattice Boltzmann scheme for magnetohydrodynamics. The antisymmetric part of the kinetic equation matches the Maxwell–Faraday equation and Ohm’s law. The symmetric part matches the hyperbolic divergence cleaning model. The discrete divergence of the magnetic field remains zero, to within round-off error, when the initial magnetic field is the discrete curl of a vector potential. We have thus constructed a mimetic or constrained transport scheme for magnetohydrodynamics.
{"title":"A magic two-relaxation-time lattice Boltzmann algorithm for magnetohydrodynamics","authors":"P. Dellar","doi":"10.3934/dcdss.2023157","DOIUrl":"https://doi.org/10.3934/dcdss.2023157","url":null,"abstract":". The two-relaxation-time collision operator in discrete kinetic theory models collisions between particles by grouping them into pairs with anti-parallel velocities. It prescribes a linear relaxation towards equilibrium with one rate for the even combination of distribution functions for each pair, and another rate for the odd combination. We reformulate this collision operator using relaxation rates for the forward-propagating and backward-propagating combinations instead. An optimal pair of relaxation rates sets the forward-propagating combination of each pair of distributions to equilibrium. Only the backward-propagating non-equilibrium distributions remain. Applying this result twice gives closed discrete equations for evolving the macroscopic variables alone across three time levels. We split the equivalent equations into a first-order system: a conservation law and a kinetic equation for the flux. All other quantities are evaluated at equilibrium. We apply this formalism to the magnetic field in a lattice Boltzmann scheme for magnetohydrodynamics. The antisymmetric part of the kinetic equation matches the Maxwell–Faraday equation and Ohm’s law. The symmetric part matches the hyperbolic divergence cleaning model. The discrete divergence of the magnetic field remains zero, to within round-off error, when the initial magnetic field is the discrete curl of a vector potential. We have thus constructed a mimetic or constrained transport scheme for magnetohydrodynamics.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"45 4 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85541214","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}
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