SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6446-6482, October 2024. Abstract. We investigate properties of the image and kernel of the Biot–Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot–Savart operator is [math]-dense in the space of square integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighborhood of the underlying plasma domain can in fact be approximated in any [math]-norm by elements of the image of the Biot–Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot–Savart operator which in particular implies that the dimension of the kernel of the Biot–Savart operator coincides with the genus of the CWS and hence turns out to be a homotopy invariant among regular domains in 3-space. Last, we provide an iterative scheme which we show converges weakly in [math]-topology to elements of the kernel of the Biot–Savart operator.
{"title":"Properties of the Biot–Savart Operator Acting on Surface Currents","authors":"Wadim Gerner","doi":"10.1137/23m1615693","DOIUrl":"https://doi.org/10.1137/23m1615693","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6446-6482, October 2024. <br/> Abstract. We investigate properties of the image and kernel of the Biot–Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot–Savart operator is [math]-dense in the space of square integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighborhood of the underlying plasma domain can in fact be approximated in any [math]-norm by elements of the image of the Biot–Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot–Savart operator which in particular implies that the dimension of the kernel of the Biot–Savart operator coincides with the genus of the CWS and hence turns out to be a homotopy invariant among regular domains in 3-space. Last, we provide an iterative scheme which we show converges weakly in [math]-topology to elements of the kernel of the Biot–Savart operator.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250643","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 6422-6445, October 2024. Abstract. The original KWC system is widely used in materials science. It was proposed in [R. Kobayashi, J. A. Warren, and W. C. Carter, Phys. D, 140 (2000), pp. 141–150] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. First, it is difficult to establish its relation to physics, in particular a variational model. Second, it lacks uniqueness. The former has been recently studied within the realm of BV theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC system. A direct relationship with variational model (as gradient flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues.
SIAM 数学分析期刊》，第 56 卷第 5 期，第 6422-6445 页，2024 年 10 月。 摘要。原始 KWC 系统广泛应用于材料科学领域。它是在[R.Kobayashi, J. A. Warren, and W. C. Carter, Phys. D, 140 (2000), pp.该模型面临两大挑战。首先，很难确定它与物理学的关系，特别是与变分模型的关系。其次，它缺乏唯一性。前者最近已在 BV 理论范畴内得到研究。后者只有在各种简化条件下才成立。本文介绍了 KWC 系统的伪抛物线版本。在不做任何不切实际的简化的情况下，建立了与变分模型（如梯度流）和唯一性的直接关系。也就是说，这是第一个在物理和数学上都有效的 KWC 系统。所提出的模型克服了众所周知的未决问题。
{"title":"Well-Posedness of a Pseudo-Parabolic KWC System in Materials Science","authors":"Harbir Antil, Daiki Mizuno, Ken Shirakawa","doi":"10.1137/24m163952x","DOIUrl":"https://doi.org/10.1137/24m163952x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6422-6445, October 2024. <br/> Abstract. The original KWC system is widely used in materials science. It was proposed in [R. Kobayashi, J. A. Warren, and W. C. Carter, Phys. D, 140 (2000), pp. 141–150] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. First, it is difficult to establish its relation to physics, in particular a variational model. Second, it lacks uniqueness. The former has been recently studied within the realm of BV theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC system. A direct relationship with variational model (as gradient flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250645","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 6398-6421, October 2024. Abstract. The paper is devoted to divergence-curl results involving a divergence free measure-valued field [math], where [math] is a signed Radon measure on [math] and [math] is a nonvanishing regular vector field in [math], and a gradient measure-valued field [math] on [math], [math]. On the one hand, in a nonperiodic framework we prove that for any open set [math] of [math], the orthogonality condition [math] in [math] implies the equality [math] in [math]. The key ingredient of the proof is based on the existence of a representative in [math] of the bounded variation function [math] in [math]. This result allows us to extend in the setting of ODE’s flows the famous Franks–Misiurewicz theorem, which claims that the Herman rotation set of any continuous two-dimensional flow on the torus [math] is a closed line segment of a line of [math] passing through [math]. Moreover, this nonperiodic divergence-curl result can be applied to a finite almost periodic bounded variation function [math] and to a finite almost periodic measure-valued field [math]. On the other hand, in the periodic case with dimension [math], assuming that [math] is absolutely continuous with respect to Lebesgue’s measure on the torus [math], we prove that if the product [math] is the zero measure on [math], so is the product of the [math]-means [math].
{"title":"A New Divergence-Curl Result for Measures. Application to the Two-Dimensional ODE’s Flow","authors":"Marc Briane, Juan Casado-Díaz","doi":"10.1137/23m1617539","DOIUrl":"https://doi.org/10.1137/23m1617539","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6398-6421, October 2024. <br/> Abstract. The paper is devoted to divergence-curl results involving a divergence free measure-valued field [math], where [math] is a signed Radon measure on [math] and [math] is a nonvanishing regular vector field in [math], and a gradient measure-valued field [math] on [math], [math]. On the one hand, in a nonperiodic framework we prove that for any open set [math] of [math], the orthogonality condition [math] in [math] implies the equality [math] in [math]. The key ingredient of the proof is based on the existence of a representative in [math] of the bounded variation function [math] in [math]. This result allows us to extend in the setting of ODE’s flows the famous Franks–Misiurewicz theorem, which claims that the Herman rotation set of any continuous two-dimensional flow on the torus [math] is a closed line segment of a line of [math] passing through [math]. Moreover, this nonperiodic divergence-curl result can be applied to a finite almost periodic bounded variation function [math] and to a finite almost periodic measure-valued field [math]. On the other hand, in the periodic case with dimension [math], assuming that [math] is absolutely continuous with respect to Lebesgue’s measure on the torus [math], we prove that if the product [math] is the zero measure on [math], so is the product of the [math]-means [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250647","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 6337-6360, October 2024. Abstract. This paper concerns subsonic jet flows from two-dimensional finitely long divergent nozzles with straight solid walls, which are governed by a free boundary problem for a quasilinear elliptic equation. It is assumed that the angle of the nozzle and the location of the inlet are fixed, while the length of the nozzle is free. For a given surrounding pressure and a given incoming mass flux, it is shown that there is a critical number not greater than [math] for the angle of the nozzle such that there exists a unique subsonic jet flow if the angle of the nozzle is less than the critical number. If this critical number is less than [math], then there is not a subsonic jet flow when the angle of the nozzle takes this critical number; furthermore, as the angle of the nozzle tends to this critical number, either the length of the nozzle tends to zero, or a sonic point will occur at the inlet. Moreover, it is shown that the subsonic jet flow tends to a uniform horizontal flow exponentially at the downstream. As to the jet, it is smooth away from the connecting point with the wall of the nozzle, and it connects the wall of the nozzle with [math] regularity for each exponent [math]. Furthermore, the jet is strictly concave to the fluid and tends to a line parallel to the symmetrical axis exponentially.
{"title":"A Free Boundary Problem in an Unbounded Domain and Subsonic Jet Flows from Divergent Nozzles","authors":"Yuanyuan Nie, Chunpeng Wang, Guanming Gai","doi":"10.1137/23m162301x","DOIUrl":"https://doi.org/10.1137/23m162301x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6337-6360, October 2024. <br/> Abstract. This paper concerns subsonic jet flows from two-dimensional finitely long divergent nozzles with straight solid walls, which are governed by a free boundary problem for a quasilinear elliptic equation. It is assumed that the angle of the nozzle and the location of the inlet are fixed, while the length of the nozzle is free. For a given surrounding pressure and a given incoming mass flux, it is shown that there is a critical number not greater than [math] for the angle of the nozzle such that there exists a unique subsonic jet flow if the angle of the nozzle is less than the critical number. If this critical number is less than [math], then there is not a subsonic jet flow when the angle of the nozzle takes this critical number; furthermore, as the angle of the nozzle tends to this critical number, either the length of the nozzle tends to zero, or a sonic point will occur at the inlet. Moreover, it is shown that the subsonic jet flow tends to a uniform horizontal flow exponentially at the downstream. As to the jet, it is smooth away from the connecting point with the wall of the nozzle, and it connects the wall of the nozzle with [math] regularity for each exponent [math]. Furthermore, the jet is strictly concave to the fluid and tends to a line parallel to the symmetrical axis exponentially.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250649","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 6361-6397, October 2024. Abstract. The ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a parameter [math], so that the correct Prandtl number can be computed in the Navier–Stokes limit. In this work, we consider steady rarefied flows arising from the evaporation and condensation process between two parallel condensed phases, which is formulated in this paper as the existence problem of stationary solutions to the ES-BGK model in a bounded interval with the mixed boundary conditions. One of the key difficulties arises in the uniform control of the temperature tensor from below. In the noncritical case [math], we utilize the property that the temperature tensor is equivalent to the temperature. In the critical case, [math], where such equivalence relation breaks down, we observe that the size of bulk velocity in the [math] direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below.
{"title":"Stationary Flows of the ES-BGK Model with the Correct Prandtl Number","authors":"Stephane Brull, Seok-Bae Yun","doi":"10.1137/23m1599628","DOIUrl":"https://doi.org/10.1137/23m1599628","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6361-6397, October 2024. <br/> Abstract. The ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a parameter [math], so that the correct Prandtl number can be computed in the Navier–Stokes limit. In this work, we consider steady rarefied flows arising from the evaporation and condensation process between two parallel condensed phases, which is formulated in this paper as the existence problem of stationary solutions to the ES-BGK model in a bounded interval with the mixed boundary conditions. One of the key difficulties arises in the uniform control of the temperature tensor from below. In the noncritical case [math], we utilize the property that the temperature tensor is equivalent to the temperature. In the critical case, [math], where such equivalence relation breaks down, we observe that the size of bulk velocity in the [math] direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250648","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}
Indranil Chowdhury, Espen R. Jakobsen, Miłosz Krupski
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6302-6336, October 2024. Abstract. We introduce a class of fully nonlinear mean field games posed in [math]. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (Lévy) processes. The main results are the existence and uniqueness of solutions under general assumptions. These results are applied to nondegenerate equations—including both local second-order and nonlocal with fractional Laplacians. Uniqueness holds under the monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.
{"title":"On Fully Nonlinear Parabolic Mean Field Games with Nonlocal and Local Diffusions","authors":"Indranil Chowdhury, Espen R. Jakobsen, Miłosz Krupski","doi":"10.1137/23m1615528","DOIUrl":"https://doi.org/10.1137/23m1615528","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6302-6336, October 2024. <br/> Abstract. We introduce a class of fully nonlinear mean field games posed in [math]. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (Lévy) processes. The main results are the existence and uniqueness of solutions under general assumptions. These results are applied to nondegenerate equations—including both local second-order and nonlocal with fractional Laplacians. Uniqueness holds under the monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203656","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 6268-6301, October 2024. Abstract. The two-dimensional stochastic Euler equations (EEs) perturbed by a linear multiplicative noise of Itô type on the bounded domain [math] have been considered in this work. Our first aim is to prove the existence of global weak (analytic) solutions for stochastic EEs when the divergence-free initial data [math], and the external forcing [math]. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if [math] and [math], we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EEs driven by linear multiplicative noise. Second, we prove the existence of a pullback stochastic weak attractor for stochastic nonautonomous EEs using the abstract theory available in the literature. Finally, we propose an abstract theory for weak asymptotic autonomy of pullback stochastic weak attractors. Then we consider the 2D stochastic EEs perturbed by a linear multiplicative noise as an example to discuss how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EEs do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EEs only. Since we are considering stochastic EEs without dissipation, all the results of this work for 2D stochastic EEs perturbed by a linear multiplicative noise are totally new.
SIAM 数学分析期刊》，第 56 卷第 5 期，第 6268-6301 页，2024 年 10 月。 摘要。本研究考虑了有界域[math]上受 Itô 型线性乘法噪声扰动的二维随机欧拉方程 (EEs)。我们的首要目标是证明在无发散初始数据[math]和外部强迫[math]条件下随机欧拉方程全局弱（解析）解的存在性。为了证明弱解的存在，我们采用了粘性消失技术。此外，如果[math]和[math]，我们确定全局弱（解析）解是唯一的。这项工作似乎是第一个讨论线性乘法噪声驱动的随机 EE 的全局弱（解析）解的存在性和唯一性的工作。其次，我们利用文献中的抽象理论证明了随机非自治 EE 的回拉随机弱吸引子的存在性。最后，我们提出了回拉随机弱吸引子弱渐近自洽性的抽象理论。然后，我们以受线性乘法噪声扰动的二维随机 EE 为例，讨论如何证明具体随机偏微分方程的弱渐近自洽性。由于 EE 不包含任何耗散项，文献中关于吸引子（确定性和随机性）的结果仅适用于耗散（或阻尼）EE。由于我们考虑的是没有耗散的随机 EE，因此本研究针对受线性乘法噪声扰动的二维随机 EE 的所有结果都是全新的。
{"title":"Theory of Weak Asymptotic Autonomy of Pullback Stochastic Weak Attractors and Its Applications to 2D Stochastic Euler Equations Driven by Multiplicative Noise","authors":"Kush Kinra, Manil T. Mohan","doi":"10.1137/24m1637878","DOIUrl":"https://doi.org/10.1137/24m1637878","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6268-6301, October 2024. <br/> Abstract. The two-dimensional stochastic Euler equations (EEs) perturbed by a linear multiplicative noise of Itô type on the bounded domain [math] have been considered in this work. Our first aim is to prove the existence of global weak (analytic) solutions for stochastic EEs when the divergence-free initial data [math], and the external forcing [math]. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if [math] and [math], we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EEs driven by linear multiplicative noise. Second, we prove the existence of a pullback stochastic weak attractor for stochastic nonautonomous EEs using the abstract theory available in the literature. Finally, we propose an abstract theory for weak asymptotic autonomy of pullback stochastic weak attractors. Then we consider the 2D stochastic EEs perturbed by a linear multiplicative noise as an example to discuss how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EEs do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EEs only. Since we are considering stochastic EEs without dissipation, all the results of this work for 2D stochastic EEs perturbed by a linear multiplicative noise are totally new.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203369","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 6232-6267, October 2024. Abstract. In this paper we will derive a nonlocal (“integral”) equation which transforms a three-dimensional acoustic transmission problem with variable coefficients, nonzero absorption, and mixed boundary conditions to a nonlocal equation on a “skeleton” of the domain [math], where “skeleton” stands for the union of the interfaces and boundaries of a Lipschitz partition of [math]. To that end, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the nonlocal skeleton equation as a direct method for the unknown Cauchy data of the solution of the original partial differential equation. We establish coercivity and continuity of the variational form of the skeleton equation based on auxiliary full space variational problems. Explicit expressions for Green’s functions is not required and all our estimates are explicit in the complex wave number.
{"title":"Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients","authors":"F. Florian, R. Hiptmair, S. A. Sauter","doi":"10.1137/23m1572106","DOIUrl":"https://doi.org/10.1137/23m1572106","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6232-6267, October 2024. <br/> Abstract. In this paper we will derive a nonlocal (“integral”) equation which transforms a three-dimensional acoustic transmission problem with variable coefficients, nonzero absorption, and mixed boundary conditions to a nonlocal equation on a “skeleton” of the domain [math], where “skeleton” stands for the union of the interfaces and boundaries of a Lipschitz partition of [math]. To that end, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the nonlocal skeleton equation as a direct method for the unknown Cauchy data of the solution of the original partial differential equation. We establish coercivity and continuity of the variational form of the skeleton equation based on auxiliary full space variational problems. Explicit expressions for Green’s functions is not required and all our estimates are explicit in the complex wave number.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203646","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 6213-6231, October 2024. Abstract. In this paper, we consider the one time-varying component regularity criteria for a local strong solution of 3-D Navier–Stokes equations. Precisely, if [math] is a piecewise [math] unit vector from [math] to [math] with finitely many jump discontinuities, we prove that if [math], then the solution [math] can be extended beyond the time [math]. Compared with the previous results J.-Y. Chemin and P. Zhang, Ann. Sci. Éc. Norm. Supér, 4 (2016), pp. 49–167 concerning one-component regularity criteria, here the unit vector [math] varies with the time variable.
SIAM 数学分析期刊》，第 56 卷第 5 期，第 6213-6231 页，2024 年 10 月。 摘要本文考虑了三维 Navier-Stokes 方程局部强解的一个时变分量正则准则。确切地说，如果[math]是一个从[math]到[math]的片状[math]单位向量，具有有限多个跳跃不连续，我们证明如果[math]，那么解[math]可以扩展到时间[math]之外。与之前的结果相比，J.-Y. Chemin 和 P. Zhang，Ann.Chemin and P. Zhang, Ann.Sci.Norm.Supér, 4 (2016), pp.
{"title":"On the One Time-Varying Component Regularity Criteria for 3-D Navier-Stokes Equations","authors":"Yanlin Liu, Ping Zhang","doi":"10.1137/23m1623124","DOIUrl":"https://doi.org/10.1137/23m1623124","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6213-6231, October 2024. <br/> Abstract. In this paper, we consider the one time-varying component regularity criteria for a local strong solution of 3-D Navier–Stokes equations. Precisely, if [math] is a piecewise [math] unit vector from [math] to [math] with finitely many jump discontinuities, we prove that if [math], then the solution [math] can be extended beyond the time [math]. Compared with the previous results J.-Y. Chemin and P. Zhang, Ann. Sci. Éc. Norm. Supér, 4 (2016), pp. 49–167 concerning one-component regularity criteria, here the unit vector [math] varies with the time variable.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203371","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}
Denis Khimin, Johannes Lankeit, Marc C. Steinbach, Thomas Wick
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6192-6212, October 2024. Abstract. The purpose of this work is the formulation of optimality conditions for phase-field optimal control problems. The forward problem is first stated as an abstract nonlinear optimization problem, and then the necessary optimality conditions are derived. The sufficient optimality conditions are also examined. The choice of suitable function spaces to ensure the regularity of the nonlinear optimization problem is a true challenge here. Afterwards the optimal control problem with a tracking type cost functional is formulated. The constraints are given by the previously derived first order optimality conditions of the forward problem. Herein regularity is proven under certain conditions and first order optimality conditions are formulated.
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