SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6121-6136, October 2024. Abstract. For the Burgers’ equation, the entropy solution becomes instantly [math] with only [math] initial data. For conservation laws with genuinely nonlinear discontinuous flux, it is well known that the [math] regularity of entropy solutions is lost. Recently, this regularity has been proved to be fractional with [math]. Moreover, for less nonlinear flux, the solution still has a fractional regularity [math]. The resulting general rule is that the regularity of entropy solutions for a discontinuous flux is less than for a smooth flux. In this paper, an optimal geometric condition on the discontinuous flux is used to recover the same regularity as for the smooth flux with the same kind of nonlinearity.
{"title":"Higher Regularity for Entropy Solutions of Conservation Laws with Geometrically Constrained Discontinuous Flux","authors":"S. S. Ghoshal, S. Junca, A. Parmar","doi":"10.1137/23m1604199","DOIUrl":"https://doi.org/10.1137/23m1604199","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6121-6136, October 2024. <br/> Abstract. For the Burgers’ equation, the entropy solution becomes instantly [math] with only [math] initial data. For conservation laws with genuinely nonlinear discontinuous flux, it is well known that the [math] regularity of entropy solutions is lost. Recently, this regularity has been proved to be fractional with [math]. Moreover, for less nonlinear flux, the solution still has a fractional regularity [math]. The resulting general rule is that the regularity of entropy solutions for a discontinuous flux is less than for a smooth flux. In this paper, an optimal geometric condition on the discontinuous flux is used to recover the same regularity as for the smooth flux with the same kind of nonlinearity.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6057-6120, October 2024. Abstract. We study a diffuse interface model describing the complex rheology and the interfacial dynamics during phase separation in a polar liquid-crystalline emulsion. More precisely, the physical systems comprises a two-phase mixture consisting of a polar liquid crystal immersed in a Newtonian fluid. Such composite material is a paradigmatic example of complex fluids arising in Soft Matter which exhibits multiscale interplay. Beyond the Ginzburg–Landau and Frank elastic energies for the concentration and the polarization, the free energy of the system is characterized by a quadratic anchoring term which tunes the orientation of the polarization at the interface. This leads to several quasi-linear nonlinear couplings in the resulting system describing the macroscopic dynamics. In this work, we establish the first mathematical results concerning the global dynamics of two-phase complex fluids with interfacial anchoring mechanism. First, we determine a set of sufficient conditions on the parameters of the system and the initial conditions which guarantee the existence of global weak solutions in two and three dimensions. Second, we show that weak solutions are unique and globally regular in the two dimensional case. Finally, we complement our analysis with some numerical simulations to display polarization and interfacial anchoring.
{"title":"Global Solutions for Two-Phase Complex Fluids with Quadratic Anchoring in Soft Matter Physics","authors":"Giulia Bevilacqua, Andrea Giorgini","doi":"10.1137/23m1608902","DOIUrl":"https://doi.org/10.1137/23m1608902","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6057-6120, October 2024. <br/> Abstract. We study a diffuse interface model describing the complex rheology and the interfacial dynamics during phase separation in a polar liquid-crystalline emulsion. More precisely, the physical systems comprises a two-phase mixture consisting of a polar liquid crystal immersed in a Newtonian fluid. Such composite material is a paradigmatic example of complex fluids arising in Soft Matter which exhibits multiscale interplay. Beyond the Ginzburg–Landau and Frank elastic energies for the concentration and the polarization, the free energy of the system is characterized by a quadratic anchoring term which tunes the orientation of the polarization at the interface. This leads to several quasi-linear nonlinear couplings in the resulting system describing the macroscopic dynamics. In this work, we establish the first mathematical results concerning the global dynamics of two-phase complex fluids with interfacial anchoring mechanism. First, we determine a set of sufficient conditions on the parameters of the system and the initial conditions which guarantee the existence of global weak solutions in two and three dimensions. Second, we show that weak solutions are unique and globally regular in the two dimensional case. Finally, we complement our analysis with some numerical simulations to display polarization and interfacial anchoring.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6137-6191, October 2024. Abstract. In this paper we explore the boundaries of damping estimates by comparing and contrasting two closely related models of combustion, the Majda and ZND models. We are especially concerned with studying the behavior of perturbations of discontinuous waves. On the one hand, we show that singularities form in the unweighted Lipschitz norm on both sides of the shock for both models, extending classical results of John in [Comm. Pure Appl. Math., 27 (1974), pp. 377–405] and Liu in [J. Differential Equations, 33 (1979), pp. 92–111] to suitable variable coefficient systems This involves adapting John’s argument to perturbations of nonconstant waves instead of perturbations of constants. On the other hand, we show instability in exponentially weighted Sobolev spaces for ZND and stability for the Majda model in similarly weighted spaces. This involves proving high order energy estimates, using convective effects and the partial decay coming from a damping term, while being careful with the boundary terms. We note that the convective effects are the origin of the instability in the weighted norm in the ZND model.
{"title":"Majda and ZND Models for Detonation: Nonlinear Stability vs. Formation of Singularities","authors":"Paul Blochas, Aric Wheeler","doi":"10.1137/23m1544945","DOIUrl":"https://doi.org/10.1137/23m1544945","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6137-6191, October 2024. <br/> Abstract. In this paper we explore the boundaries of damping estimates by comparing and contrasting two closely related models of combustion, the Majda and ZND models. We are especially concerned with studying the behavior of perturbations of discontinuous waves. On the one hand, we show that singularities form in the unweighted Lipschitz norm on both sides of the shock for both models, extending classical results of John in [Comm. Pure Appl. Math., 27 (1974), pp. 377–405] and Liu in [J. Differential Equations, 33 (1979), pp. 92–111] to suitable variable coefficient systems This involves adapting John’s argument to perturbations of nonconstant waves instead of perturbations of constants. On the other hand, we show instability in exponentially weighted Sobolev spaces for ZND and stability for the Majda model in similarly weighted spaces. This involves proving high order energy estimates, using convective effects and the partial decay coming from a damping term, while being careful with the boundary terms. We note that the convective effects are the origin of the instability in the weighted norm in the ZND model.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5840-5880, October 2024. Abstract. Positron emission tomography (PET) is a classical imaging technique to reconstruct the mass distribution of a radioactive material. If the mass distribution is static, this essentially leads to inversion of the X-ray transform. However, if the mass distribution changes temporally, the measurement signals received over time (the so-called listmode data) belong to different spatial configurations. We suggest and analyze a Bayesian approach to solve this dynamic inverse problem that is based on optimal transport regularization of the temporally changing mass distribution. Our focus lies on a rigorous derivation of the Bayesian model and the analysis of its properties, treating both the continuous as well as the discrete (finitely many detectors and time binning) setting.
{"title":"A Bayesian Model for Dynamic Mass Reconstruction from PET Listmode Data","authors":"Marco Mauritz, Bernhard Schmitzer, Benedikt Wirth","doi":"10.1137/23m161923x","DOIUrl":"https://doi.org/10.1137/23m161923x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5840-5880, October 2024. <br/> Abstract. Positron emission tomography (PET) is a classical imaging technique to reconstruct the mass distribution of a radioactive material. If the mass distribution is static, this essentially leads to inversion of the X-ray transform. However, if the mass distribution changes temporally, the measurement signals received over time (the so-called listmode data) belong to different spatial configurations. We suggest and analyze a Bayesian approach to solve this dynamic inverse problem that is based on optimal transport regularization of the temporally changing mass distribution. Our focus lies on a rigorous derivation of the Bayesian model and the analysis of its properties, treating both the continuous as well as the discrete (finitely many detectors and time binning) setting.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Roberta Bianchini, Lars Eric Hientzsch, Felice Iandoli
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5915-5968, October 2024. Abstract. We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in [math] without boundary, building upon the method that Elgindi and Shikh Khalil [Strong Ill-Posedness in [math] for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small [math] size, for which the horizontal density gradient [math] has a strong [math]-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi’s decomposition of the Biot–Savart law, we apply our method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with vorticity uniformly bounded and small in [math] provides a solution whose gradient of the swirl has a strong [math]-norm inflation in infinitesimal time. The norm inflation is quantified from below by an explicit lower bound which depends on time and the size of the data and is valid for small times.
{"title":"Strong Ill-Posedness in [math] of the 2D Boussinesq Equations in Vorticity Form and Application to the 3D Axisymmetric Euler Equations","authors":"Roberta Bianchini, Lars Eric Hientzsch, Felice Iandoli","doi":"10.1137/23m159384x","DOIUrl":"https://doi.org/10.1137/23m159384x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5915-5968, October 2024. <br/> Abstract. We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in [math] without boundary, building upon the method that Elgindi and Shikh Khalil [Strong Ill-Posedness in [math] for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small [math] size, for which the horizontal density gradient [math] has a strong [math]-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi’s decomposition of the Biot–Savart law, we apply our method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with vorticity uniformly bounded and small in [math] provides a solution whose gradient of the swirl has a strong [math]-norm inflation in infinitesimal time. The norm inflation is quantified from below by an explicit lower bound which depends on time and the size of the data and is valid for small times.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5881-5914, October 2024. Abstract. Geophysicists have studied three-dimensional quasi-geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large timescale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in [math] with an equation on the boundary satisfied by the trace, where [math] is either a two-dimensional torus or a bounded convex domain in [math]. In this paper, we show the existence of global in time weak solutions to a family of singular three-dimensional quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary.
{"title":"Global in Time Weak Solutions to Singular Three-Dimensional Quasi-Geostrophic Systems","authors":"Yiran Hu","doi":"10.1137/23m1552917","DOIUrl":"https://doi.org/10.1137/23m1552917","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5881-5914, October 2024. <br/> Abstract. Geophysicists have studied three-dimensional quasi-geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large timescale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in [math] with an equation on the boundary satisfied by the trace, where [math] is either a two-dimensional torus or a bounded convex domain in [math]. In this paper, we show the existence of global in time weak solutions to a family of singular three-dimensional quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maarten V. de Hoop, Masato Kimura, Ching-Lung Lin, Gen Nakamura
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5782-5806, October 2024. Abstract. In the theory of viscoelasticity, an important class of models admits a representation in terms of springs and dashpots. Widely used members of this class are the Maxwell model and its extended version. This paper concerns resolvent estimates for the system of equations for the anisotropic, extended Maxwell model (EMM) and its marginal model; special attention is paid to the introduction of augmented variables. This leads to the augmented system that will also be referred to as the “original” system. A reduced system is then formed which encodes essentially the EMM; it is a closed system with respect to the particle velocity and the difference between the elastic and viscous strains. Based on resolvent estimates, it is shown that the original and reduced systems generate [math]-groups and the reduced system generates a [math]-semigroup of contraction. Naturally, the EMM can be written in an integro-differential form with a relaxation tensor. However, there is a difference between the original and integro-differential systems, in general, with consequences for whether their solutions generate semigroups or not. Finally, an energy estimate is obtained for the reduced system, and it is proven that its solutions decay exponentially as time tends to infinity. The limiting amplitude principle follows readily from these two results.
{"title":"Resolvent Estimates for Viscoelastic Systems of Extended Maxwell Type and Their Applications","authors":"Maarten V. de Hoop, Masato Kimura, Ching-Lung Lin, Gen Nakamura","doi":"10.1137/23m1592456","DOIUrl":"https://doi.org/10.1137/23m1592456","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5782-5806, October 2024. <br/> Abstract. In the theory of viscoelasticity, an important class of models admits a representation in terms of springs and dashpots. Widely used members of this class are the Maxwell model and its extended version. This paper concerns resolvent estimates for the system of equations for the anisotropic, extended Maxwell model (EMM) and its marginal model; special attention is paid to the introduction of augmented variables. This leads to the augmented system that will also be referred to as the “original” system. A reduced system is then formed which encodes essentially the EMM; it is a closed system with respect to the particle velocity and the difference between the elastic and viscous strains. Based on resolvent estimates, it is shown that the original and reduced systems generate [math]-groups and the reduced system generates a [math]-semigroup of contraction. Naturally, the EMM can be written in an integro-differential form with a relaxation tensor. However, there is a difference between the original and integro-differential systems, in general, with consequences for whether their solutions generate semigroups or not. Finally, an energy estimate is obtained for the reduced system, and it is proven that its solutions decay exponentially as time tends to infinity. The limiting amplitude principle follows readily from these two results.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5995-6024, October 2024. Abstract. Reactions with enzymes are critical in biochemistry, where the enzymes act as a catalyst in the process. One of the most widely used mechanisms for modeling enzyme-catalyzed reactions is the Michaelis–Menten (MM) kinetics. On the ODE level, i.e., when concentrations are only time-dependent, this kinetics can be rigorously derived from mass action law using quasi-steady-state approximation. This issue in the PDE setting, for instance, when molecular diffusion is taken into account, is considerably more challenging, and only formal derivations have been established. In this paper, we prove this derivation rigorously and obtain MM kinetics in the presence of slow spatial diffusion. In particular, we show in this case that, in general, the reduced problem is a cross-diffusion-reaction system. Our proof is based on an improved duality method, heat regularization, and a suitable modified energy function. To the best of our knowledge, this work provides the first rigorous derivation of MM kinetics from mass action kinetics in the PDE setting.
{"title":"Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion","authors":"Bao Quoc Tang, Bao-Ngoc Tran","doi":"10.1137/23m1579406","DOIUrl":"https://doi.org/10.1137/23m1579406","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5995-6024, October 2024. <br/> Abstract. Reactions with enzymes are critical in biochemistry, where the enzymes act as a catalyst in the process. One of the most widely used mechanisms for modeling enzyme-catalyzed reactions is the Michaelis–Menten (MM) kinetics. On the ODE level, i.e., when concentrations are only time-dependent, this kinetics can be rigorously derived from mass action law using quasi-steady-state approximation. This issue in the PDE setting, for instance, when molecular diffusion is taken into account, is considerably more challenging, and only formal derivations have been established. In this paper, we prove this derivation rigorously and obtain MM kinetics in the presence of slow spatial diffusion. In particular, we show in this case that, in general, the reduced problem is a cross-diffusion-reaction system. Our proof is based on an improved duality method, heat regularization, and a suitable modified energy function. To the best of our knowledge, this work provides the first rigorous derivation of MM kinetics from mass action kinetics in the PDE setting.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5969-5994, October 2024. Abstract. In this paper, we consider the measure valued solution to the homogeneous Boltzmann equation with Debye–Yukawa potential. First, for the case of the true Debye–Yukawa potential, we prove global in time existence of a solution with properties such as gain of moments. And then we consider the Maxwellian molecule of Deybe–Yukawa type potential. By introducing a new coercivity estimate, we show that the solution [math] for any [math], as long as the initial data [math] is not orthogonal to two translations of the measure itself.
{"title":"Existence and Smoothing Effect of Measure Valued Solution to the Homogeneous Boltzmann Equation with Debye–Yukawa Potential","authors":"Shuaikun Wang, Tong Yang","doi":"10.1137/23m1562123","DOIUrl":"https://doi.org/10.1137/23m1562123","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5969-5994, October 2024. <br/> Abstract. In this paper, we consider the measure valued solution to the homogeneous Boltzmann equation with Debye–Yukawa potential. First, for the case of the true Debye–Yukawa potential, we prove global in time existence of a solution with properties such as gain of moments. And then we consider the Maxwellian molecule of Deybe–Yukawa type potential. By introducing a new coercivity estimate, we show that the solution [math] for any [math], as long as the initial data [math] is not orthogonal to two translations of the measure itself.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6025-6056, October 2024. Abstract. The paper is concerned with the monotonicity of the principal eigenvalues with respect to diffusion rate for two classes of non-self-adjoint operators: time-periodic parabolic operators and elliptic operators with shear flow. These operators behave similarly to some averaged self-adjoint elliptic operators when the frequency or flow amplitude, referred to as advection rate, is sufficiently large. It was conjectured in [S. Liu and Y. Lou, J. Funct. Anal., 282 (2022), 109338] that the principal eigenvalues are monotone in diffusion rate for large advection, similarly to those self-adjoint elliptic operators. We provide some counterexamples to the conjecture by establishing the high order expansion of the principal eigenvalues for sufficiently large diffusion and advection rates.
SIAM 数学分析期刊》，第 56 卷第 5 期，第 6025-6056 页，2024 年 10 月。 摘要本文关注两类非自相加算子的主特征值关于扩散率的单调性：时周期抛物线算子和具有剪切流的椭圆算子。当频率或流动振幅（称为平流率）足够大时，这些算子的行为类似于某些平均自相交椭圆算子。S. Liu 和 Y. Lou，J. Funct. Anal.，282 (2022), 109338]中猜想，在大平流情况下，主特征值在扩散率上是单调的，这与那些自相关椭圆算子类似。我们通过建立主特征值在足够大的扩散率和平流率下的高阶展开，为猜想提供了一些反例。
{"title":"Nonmonotonicity of Principal Eigenvalues in Diffusion Rate for Some Non-Self-Adjoint Operators with Large Advection","authors":"Shuang Liu","doi":"10.1137/23m1603947","DOIUrl":"https://doi.org/10.1137/23m1603947","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6025-6056, October 2024. <br/> Abstract. The paper is concerned with the monotonicity of the principal eigenvalues with respect to diffusion rate for two classes of non-self-adjoint operators: time-periodic parabolic operators and elliptic operators with shear flow. These operators behave similarly to some averaged self-adjoint elliptic operators when the frequency or flow amplitude, referred to as advection rate, is sufficiently large. It was conjectured in [S. Liu and Y. Lou, J. Funct. Anal., 282 (2022), 109338] that the principal eigenvalues are monotone in diffusion rate for large advection, similarly to those self-adjoint elliptic operators. We provide some counterexamples to the conjecture by establishing the high order expansion of the principal eigenvalues for sufficiently large diffusion and advection rates.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号