The existence of solitary wave solutions of the one-dimensional version of the fractional nonlinear Schrödinger (fNLS) equation was analyzed by the authors in a previous work. In this paper, the asymptotic decay of the solitary waves is analyzed. From the formulation of the differential system for the wave profiles as a convolution, these are shown to decay algebraically to zero at infinity, with an order which depends on the parameter determining the fractional order of the equation. Some numerical experiments illustrate the result.
{"title":"Asymptotic Decay of Solitary Wave Solutions of the Fractional Nonlinear Schrödinger Equation","authors":"Angel Durán, Nuria Reguera","doi":"10.1111/sapm.70149","DOIUrl":"https://doi.org/10.1111/sapm.70149","url":null,"abstract":"<div>\u0000 \u0000 <p>The existence of solitary wave solutions of the one-dimensional version of the fractional nonlinear Schrödinger (fNLS) equation was analyzed by the authors in a previous work. In this paper, the asymptotic decay of the solitary waves is analyzed. From the formulation of the differential system for the wave profiles as a convolution, these are shown to decay algebraically to zero at infinity, with an order which depends on the parameter determining the fractional order of the equation. Some numerical experiments illustrate the result.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145626602","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}
In this paper, we study the eigenvalue problem for cooperative systems where the eigenvalue parameter appears on both the equation and the boundary. By utilizing a series of one-parameter eigenvalue problems, we give a sufficient condition for the existence of the positive eigenvalue, which corresponds to the positive eigenfunction, and prove that it is unique when the system is symmetric. Then, we apply the theoretical result to investigate the existence and stability of non-constant solutions for a general reaction-diffusion model with nonlinear boundary conditions. In addition, the influence of nonlinear boundary conditions on the long-time behavior of the solution is illustrated by numerical simulations.
{"title":"The Principal Eigenvalue of Cooperative Systems With Applications to a Model of Nonlinear Boundary Conditions","authors":"Suriguga, Jianhua Wu, Lei Zhang","doi":"10.1111/sapm.70148","DOIUrl":"https://doi.org/10.1111/sapm.70148","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we study the eigenvalue problem for cooperative systems where the eigenvalue parameter appears on both the equation and the boundary. By utilizing a series of one-parameter eigenvalue problems, we give a sufficient condition for the existence of the positive eigenvalue, which corresponds to the positive eigenfunction, and prove that it is unique when the system is symmetric. Then, we apply the theoretical result to investigate the existence and stability of non-constant solutions for a general reaction-diffusion model with nonlinear boundary conditions. In addition, the influence of nonlinear boundary conditions on the long-time behavior of the solution is illustrated by numerical simulations.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625970","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}
In this paper, we analyze the dynamics of a pattern-forming system close to simultaneous Turing and Turing–Hopf instabilities, which have a 1:1 spatial resonance, that is, they have the same critical wave number. For this, we consider a system of coupled Swift–Hohenberg equations with dispersive terms and general, smooth nonlinearities. Close to the onset of instability, we derive a system of two coupled complex Ginzburg–Landau equations with a singular advection term as amplitude equations and justify the approximation by providing error estimates. We then construct space-time periodic solutions to the amplitude equations, as well as fast-traveling front solutions, which connect different space-time periodic states. This yields the existence of solutions to the pattern-forming system on a finite, but long time interval, which model the spatial transition between different patterns. The construction is based on geometric singular perturbation theory exploiting the fast traveling speed of the fronts. Finally, we construct global, spatially periodic solutions to the pattern-forming system by using center manifold reduction, normal form theory, and a variant of singular perturbation theory to handle fast oscillatory higher order terms.
{"title":"Pattern Formation and Nonlinear Waves Close to a 1:1 Resonant Turing and Turing–Hopf Instability","authors":"Bastian Hilder, Christian Kuehn","doi":"10.1111/sapm.70140","DOIUrl":"https://doi.org/10.1111/sapm.70140","url":null,"abstract":"<p>In this paper, we analyze the dynamics of a pattern-forming system close to simultaneous Turing and Turing–Hopf instabilities, which have a 1:1 spatial resonance, that is, they have the same critical wave number. For this, we consider a system of coupled Swift–Hohenberg equations with dispersive terms and general, smooth nonlinearities. Close to the onset of instability, we derive a system of two coupled complex Ginzburg–Landau equations with a singular advection term as amplitude equations and justify the approximation by providing error estimates. We then construct space-time periodic solutions to the amplitude equations, as well as fast-traveling front solutions, which connect different space-time periodic states. This yields the existence of solutions to the pattern-forming system on a finite, but long time interval, which model the spatial transition between different patterns. The construction is based on geometric singular perturbation theory exploiting the fast traveling speed of the fronts. Finally, we construct global, spatially periodic solutions to the pattern-forming system by using center manifold reduction, normal form theory, and a variant of singular perturbation theory to handle fast oscillatory higher order terms.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70140","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145572555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we concentrate on the high-order rogue waves of the Davey–Stewartson I equation and investigate their dynamics in great detail. The approximate trajectories of curvy rogue waves in the intermediate stage and certain lumps at large time are determined by a special polynomial which is expressed by some Schur polynomials and their derivatives. Besides, the number of lumps at large time is illustrated by using a positive integer partition and its corresponding Young diagram. The true and estimated results show excellent agreement.
{"title":"Dynamics of the High-Order Rogue Waves in the Davey–Stewartson I Equation","authors":"Wei Wei, Yuqing Yang, Xiaoyu Chang, Lijuan Guo","doi":"10.1111/sapm.70143","DOIUrl":"https://doi.org/10.1111/sapm.70143","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we concentrate on the high-order rogue waves of the Davey–Stewartson I equation and investigate their dynamics in great detail. The approximate trajectories of curvy rogue waves in the intermediate stage and certain lumps at large time are determined by a <i>special</i> polynomial which is expressed by some Schur polynomials and their derivatives. Besides, the number of lumps at large time is illustrated by using a positive integer partition and its corresponding Young diagram. The true and estimated results show excellent agreement.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145572554","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}
In this paper, we revisit the eigenvalue problem of the one-dimensional Schrödinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr–Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues and show that the Bohr–Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of . We find that whereas the small eigenvalues are smooth functions of , the large ones are smooth functions of
本文重新研究光滑单井势一维Schrödinger方程的特征值问题。特别地，我们提供了玻尔-索默菲尔德量子化公式的一个新的解释。我们的结果的一个新方面，是基于作者最近对基于动力系统方法的转折点问题的工作，是我们涵盖了所有特征值E∈[0，O (1)] $Ein [0,mathcal {O}(1)]$并证明玻尔-索默菲尔德量化公式近似于所有这些特征值（在某种意义上是精确的）。同时，我们给出了特征值作为ε $epsilon$函数的严格光滑性表述。我们发现小特征值E = O (ε) $E=mathcal {O}(epsilon)$是ε $epsilon$的光滑函数，较大的E = O (1) $E=mathcal {O}(1)$是n ε∈[c1]的光滑函数，C 2], 0 ＆lt; C 1 ＆lt; C 2 ＆lt;∞$nepsilon in [c_1,c_2],,0<c_1<c_2<infty$，且0≤ε 1 / 3≪1 $0le epsilon ^{1/3}ll 1$；这里n∈n0 $nin mathbb {N}_0$是特征值的索引。
{"title":"A Dynamical Systems Approach to WKB-Methods: The Eigenvalue Problem for a Single Well Potential","authors":"K. Uldall Kristiansen, P. Szmolyan","doi":"10.1111/sapm.70141","DOIUrl":"https://doi.org/10.1111/sapm.70141","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we revisit the eigenvalue problem of the one-dimensional Schrödinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr–Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>∈</mo>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$Ein [0,mathcal {O}(1)]$</annotation>\u0000 </semantics></math> and show that the Bohr–Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$epsilon$</annotation>\u0000 </semantics></math>. We find that whereas the small eigenvalues <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>=</mo>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mi>ε</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$E=mathcal {O}(epsilon)$</annotation>\u0000 </semantics></math> are smooth functions of <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$epsilon$</annotation>\u0000 </semantics></math>, the large ones <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>=</mo>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$E=mathcal {O}(1)$</annotation>\u0000 </semantics></math> are smooth functions of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mi>ε</mi>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <msub>\u0000 <mi>c</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>c</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mn>0</mn>\u0000 <mo><</mo>\u0000 <msub>\u0000 <mi>c</mi>\u0000 ","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145572472","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}
Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in norm can be derived.
{"title":"A Seamless Local-Nonlocal Coupling Diffusion Model With H1 Vanishing Nonlocality Convergence","authors":"Yanzun Meng, Zuoqiang Shi","doi":"10.1111/sapm.70147","DOIUrl":"https://doi.org/10.1111/sapm.70147","url":null,"abstract":"<div>\u0000 \u0000 <p>Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$H^1$</annotation>\u0000 </semantics></math> norm can be derived.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145572518","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}
Various biological cells secrete diffusing chemical compounds into their environment for communication purposes. Secretion usually takes place over the cell membrane in a spatially heterogeneous manner. Mathematical models of these processes will be part of more elaborate models, for example, of the movement of immune cells that react to cytokines in their environment. Here, we compare two approaches to modelling of the secretion–diffusion process of signaling compounds. The first is the so-called spatial exclusion model, in which the intracellular space is excluded from consideration and the computational space is the extracellular environment. The second consists of point source models, where the secreting cell is replaced by one or more nonspatial point sources or sinks, using—mathematically—Dirac delta distributions. We propose a multi-Dirac approach and provide explicit expressions for the intensities of the Dirac distributions. We show that two to three well-positioned Dirac points suffice to approximate well a temporally constant but spatially heterogeneous flux distribution of compound over the cell membrane, for a wide range of variation in flux density and diffusivity. The multi-Dirac approach is compared to a single-Dirac approach that was studied in previous work. Moreover, an explicit Green's function approach is introduced that has significant benefits in circumventing numerical instability that may occur when the Dirac sources have high intensities.
{"title":"Approximating a Spatially-Heterogeneously Mass-Emitting Object by Multiple Point Sources in a Diffusion Model","authors":"Qiyao Peng, Sander C. Hille","doi":"10.1111/sapm.70131","DOIUrl":"https://doi.org/10.1111/sapm.70131","url":null,"abstract":"<p>Various biological cells secrete diffusing chemical compounds into their environment for communication purposes. Secretion usually takes place over the cell membrane in a spatially heterogeneous manner. Mathematical models of these processes will be part of more elaborate models, for example, of the movement of immune cells that react to cytokines in their environment. Here, we compare two approaches to modelling of the secretion–diffusion process of signaling compounds. The first is the so-called <i>spatial exclusion model</i>, in which the intracellular space is excluded from consideration and the computational space is the extracellular environment. The second consists of <i>point source models</i>, where the secreting cell is replaced by one or more nonspatial point sources or sinks, using—mathematically—Dirac delta distributions. We propose a multi-Dirac approach and provide explicit expressions for the intensities of the Dirac distributions. We show that two to three well-positioned Dirac points suffice to approximate well a temporally constant but spatially heterogeneous flux distribution of compound over the cell membrane, for a wide range of variation in flux density and diffusivity. The multi-Dirac approach is compared to a single-Dirac approach that was studied in previous work. Moreover, an explicit Green's function approach is introduced that has significant benefits in circumventing numerical instability that may occur when the Dirac sources have high intensities.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70131","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145572428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layered resonator has prompted the exploration of multi-layered metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called “resonators”, and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layered high-contrast structures. First, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layered structures within general geometry. Then, we prove the existence of subwavelength resonances via the Gohberg–Sigal theory, which generalizes the celebrated Minnaert resonances in single-layered structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results for different nested resonators are presented to corroborate the theoretical findings.
{"title":"Mathematical Theory on Multi-Layered High-Contrast Acoustic Subwavelength Resonators","authors":"Youjun Deng, Lingzheng Kong, Hongjie Li, Hongyu Liu, Liyan Zhu","doi":"10.1111/sapm.70145","DOIUrl":"https://doi.org/10.1111/sapm.70145","url":null,"abstract":"<div>\u0000 \u0000 <p>Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layered resonator has prompted the exploration of multi-layered metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called “resonators”, and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layered high-contrast structures. First, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layered structures within general geometry. Then, we prove the existence of subwavelength resonances via the Gohberg–Sigal theory, which generalizes the celebrated Minnaert resonances in single-layered structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results for different nested resonators are presented to corroborate the theoretical findings.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145521697","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}
<p>A reduced-dimensional asymptotic modeling approach is presented for the analysis of two-phase flow in a thin cylinder with an aperture of order <span></span><math> <semantics> <mrow> <mi>O</mi> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow> <annotation>$mathcal {O}(varepsilon)$</annotation> </semantics></math>, where <span></span><math> <semantics> <mi>ε</mi> <annotation>$varepsilon$</annotation> </semantics></math> is a small positive parameter. We consider a nonlinear Muskat–Leverett two-phase flow model expressed in terms of a fractional flow formulation and Darcy's law, with saturation and reduced pressure as unknown. The given flow seeps through the lateral surface of the cylinder. This exchange process leads to a nonhomogeneous Neumann boundary condition with an intensity factor <span></span><math> <semantics> <msup> <mi>ε</mi> <mi>α</mi> </msup> <annotation>$varepsilon ^alpha$</annotation> </semantics></math> <span></span><math> <semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>≥</mo> <mn>1</mn> <mo>)</mo> </mrow> <annotation>$(alpha ge 1)$</annotation> </semantics></math> that controls mass transport. Furthermore, the absolute permeability tensor comprises the intensity coefficient <span></span><math> <semantics> <msup> <mi>ε</mi> <mi>β</mi> </msup> <annotation>$varepsilon ^beta$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mi>R</mi> </mrow> <annotation>$beta in mathbb {R}$</annotation> </semantics></math>, in the transverse direction. The asymptotic behavior of the solution is studied as <span></span><math> <semantics> <mrow> <mi>ε</mi> <mo>→</mo> <mn>0</mn> </mrow> <annotation>$varepsilon rightarrow 0$</annotation> </semantics></math>, that is, when the thin cylinder shrinks into an interval. Two qualitatively distinct cases were discovered in the asymptotic behavior of the solution: <span></span><math> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1<
{"title":"Muskat–Leverett Two-Phase Flow in Thin Cylindric Porous Media: Asymptotic Approach","authors":"Taras Mel'nyk, Christian Rohde","doi":"10.1111/sapm.70137","DOIUrl":"https://doi.org/10.1111/sapm.70137","url":null,"abstract":"<p>A reduced-dimensional asymptotic modeling approach is presented for the analysis of two-phase flow in a thin cylinder with an aperture of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mi>ε</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {O}(varepsilon)$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math> is a small positive parameter. We consider a nonlinear Muskat–Leverett two-phase flow model expressed in terms of a fractional flow formulation and Darcy's law, with saturation and reduced pressure as unknown. The given flow seeps through the lateral surface of the cylinder. This exchange process leads to a nonhomogeneous Neumann boundary condition with an intensity factor <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ε</mi>\u0000 <mi>α</mi>\u0000 </msup>\u0000 <annotation>$varepsilon ^alpha$</annotation>\u0000 </semantics></math> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>α</mi>\u0000 <mo>≥</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(alpha ge 1)$</annotation>\u0000 </semantics></math> that controls mass transport. Furthermore, the absolute permeability tensor comprises the intensity coefficient <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ε</mi>\u0000 <mi>β</mi>\u0000 </msup>\u0000 <annotation>$varepsilon ^beta$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mo>∈</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$beta in mathbb {R}$</annotation>\u0000 </semantics></math>, in the transverse direction. The asymptotic behavior of the solution is studied as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varepsilon rightarrow 0$</annotation>\u0000 </semantics></math>, that is, when the thin cylinder shrinks into an interval. Two qualitatively distinct cases were discovered in the asymptotic behavior of the solution: <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>=</mo>\u0000 <mn>1<","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70137","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145521857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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