Pub Date : 2024-05-21DOI: 10.4310/dpde.2024.v21.n3.a3
Q-Heung Choi, Tacksun Jung
We deal with a family of the fractional N-Laplacian heat flows with variable exponent time-derivative on the Orlicz-Sobolev spaces. We get the maximum principle for these problems. We use the approximating method to get this result: We first show existence of a unique family of the approximating weak solutions from the variable exponent difference fractional N-Laplacian problems. We next show the maximum principle for the family of the approximating weak solution from the variable exponent difference fractional N-Laplacian problem, show the convergence of a family of the approximating weak solutions to the limits, and then obtain the maximum principle for the weak solution of a family of the fractional N-Laplacian heat flows with the variable exponent time-derivative on the Orlicz-Sobolev spaces.
{"title":"Maximum principle for the fractional N-Laplacian flow","authors":"Q-Heung Choi, Tacksun Jung","doi":"10.4310/dpde.2024.v21.n3.a3","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n3.a3","url":null,"abstract":"We deal with a family of the fractional N-Laplacian heat flows with variable exponent time-derivative on the Orlicz-Sobolev spaces. We get the maximum principle for these problems. We use the approximating method to get this result: We first show existence of a unique family of the approximating weak solutions from the variable exponent difference fractional N-Laplacian problems. We next show the maximum principle for the family of the approximating weak solution from the variable exponent difference fractional N-Laplacian problem, show the convergence of a family of the approximating weak solutions to the limits, and then obtain the maximum principle for the weak solution of a family of the fractional N-Laplacian heat flows with the variable exponent time-derivative on the Orlicz-Sobolev spaces.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.4310/dpde.2024.v21.n3.a4
Kaige Hao, Yeping Li, Rong Yin
In this paper, the low Mach number limit for the three-dimensional full compressible Navier-Stokes-Korteweg equations with general initial data is rigorously justified within the framework of local smooth solution. Under the assumption of large temperature variations, we first obtain the uniform-in- Mach-number estimates of the solutions in a $varepsilon$-weighted Sobolev space, which establishes the local existence theorem of the three-dimensional full compressible Navier-Stokes-Korteweg equations on a finite time interval independent of Mach number. Then, the low mach limit is proved by combining the uniform estimates and a strong convergence theorem of the solution for the acoustic wave equations. This result improves that of [K.-J. Sha and Y.-P. Li, Z. Angew. Math. Phys., 70(2019), 169] for well-prepared initial data.
本文在局部平稳解的框架内严格论证了具有一般初始数据的三维全可压缩纳维-斯托克斯-科特韦格方程的低马赫数极限。在温度变化较大的假设下，我们首先得到了$varepsilon$加权Sobolev空间中解的均匀马赫数估计值，从而建立了三维全可压缩Navier-Stokes-Korteweg方程在与马赫数无关的有限时间区间上的局部存在定理。然后，结合声波方程解的均匀估计和强收敛定理，证明了低马赫极限。这一结果改进了 [K.-J. Sha and Y.-P. Li, Z. Angew. Math. Phys.
{"title":"Low Mach number limit of the full compressibleNavier-Stokes-Korteweg equations with general initial data","authors":"Kaige Hao, Yeping Li, Rong Yin","doi":"10.4310/dpde.2024.v21.n3.a4","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n3.a4","url":null,"abstract":"In this paper, the low Mach number limit for the three-dimensional full compressible Navier-Stokes-Korteweg equations with general initial data is rigorously justified within the framework of local smooth solution. Under the assumption of large temperature variations, we first obtain the uniform-in- Mach-number estimates of the solutions in a $varepsilon$-weighted Sobolev space, which establishes the local existence theorem of the three-dimensional full compressible Navier-Stokes-Korteweg equations on a finite time interval independent of Mach number. Then, the low mach limit is proved by combining the uniform estimates and a strong convergence theorem of the solution for the acoustic wave equations. This result improves that of [K.-J. Sha and Y.-P. Li, Z. Angew. Math. Phys., 70(2019), 169] for well-prepared initial data.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.4310/dpde.2024.v21.n3.a1
Yuri N. Skiba
The linear and nonlinear stability of modons and Wu-Verkley waves, which are weak solutions of the barotropic vorticity equation on a rotating sphere, are analyzed. Necessary conditions for normal mode instability are obtained, the growth rate of unstable modes is estimated, and the orthogonality of unstable modes to the basic flow is shown. The Liapunov instability of dipole modons in the norm associated with enstrophy is proven.
{"title":"Stability of a class of solutions of the barotropic vorticity equation on a sphereequation on a sphere","authors":"Yuri N. Skiba","doi":"10.4310/dpde.2024.v21.n3.a1","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n3.a1","url":null,"abstract":"The linear and nonlinear stability of modons and Wu-Verkley waves, which are weak solutions of the barotropic vorticity equation on a rotating sphere, are analyzed. Necessary conditions for normal mode instability are obtained, the growth rate of unstable modes is estimated, and the orthogonality of unstable modes to the basic flow is shown. The Liapunov instability of dipole modons in the norm associated with enstrophy is proven.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.4310/dpde.2024.v21.n3.a2
Jean-Pierre Magnot, Enrique G. Reyes
We establish a rigorous link between infinite-dimensional regular Frolicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a (parameter-depending) version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator of our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase’s work on the KP hierarchy, we prove a group factorization theorem for this group of Fourier integral operators.
{"title":"On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms","authors":"Jean-Pierre Magnot, Enrique G. Reyes","doi":"10.4310/dpde.2024.v21.n3.a2","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n3.a2","url":null,"abstract":"We establish a rigorous link between infinite-dimensional regular Frolicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a (parameter-depending) version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator of our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase’s work on the KP hierarchy, we prove a group factorization theorem for this group of Fourier integral operators.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.4310/dpde.2024.v21.n1.a4
Ling Zhou, Chun-Lei Tang
We show the global well-posedness to the three-dimensional (3D) Cauchy problem of nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in $mathbb{R}^3$ provided that the initial mass is sufficiently small. Moreover, we also obtain that the gradients of velocity and micro-rotational velocity converge exponentially to zero in $H^1$ as time goes to infinity. Our analysis relies heavily on delicate energy estimates and the structural characteristic of the system under consideration. In particular, the initial velocity and micro-rotational velocity could be arbitrarily large.
{"title":"Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity","authors":"Ling Zhou, Chun-Lei Tang","doi":"10.4310/dpde.2024.v21.n1.a4","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n1.a4","url":null,"abstract":"We show the global well-posedness to the three-dimensional (3D) Cauchy problem of nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in $mathbb{R}^3$ provided that the initial mass is sufficiently small. Moreover, we also obtain that the gradients of velocity and micro-rotational velocity converge exponentially to zero in $H^1$ as time goes to infinity. Our analysis relies heavily on delicate energy estimates and <i>the structural characteristic of the system under consideration</i>. In particular, the initial velocity and micro-rotational velocity could be arbitrarily large.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.4310/dpde.2024.v21.n1.a3
Pan Zheng
and nonlocal growth term[begin{cases}u_t = Delta u - chi nabla cdot (u^m nabla v) + u Biggl( a_0 - a_1 u^alpha + a_2 displaystyle int_Omega u^sigma dx Biggr) & (x, t) in Omega times (0,infty) ; , 0=Delta - v + u^gamma , & (x, t) in Omega times (0,infty) ; , end{cases}]under homogeneous Neumann boundary conditions in a smoothly bounded domain $Omega subset mathbb{R}^n (n geq 1)$, where $chi in mathbb{R}, m, gamma geq 1$ and $a_0, a_1, a_2, alpha gt 0$. • When $chi gt 0$, the solution of the above system is global and uniformly bounded, if the parameters satisfy certain suitable assumptions. • When $chi gt 0$, the system possesses a globally bounded classical solution, provided that $a_1 gt a_2 lvert Omega rvert$. These results indicate that the repulsive mechanism plays a crucial role in ensuring the global boundedness of solutions. In addition, the paper derives the large time behavior of globally bounded solutions for the chemo-attractive or chemo-repulsive system by constructing energy functionals.
和光滑有界域$Omega subset mathbb{R}^n (n geq 1)$上齐次Neumann边界条件下的非局部增长项[begin{cases}u_t = Delta u - chi nabla cdot (u^m nabla v) + u Biggl( a_0 - a_1 u^alpha + a_2 displaystyle int_Omega u^sigma dx Biggr) & (x, t) in Omega times (0,infty) ; , 0=Delta - v + u^gamma , & (x, t) in Omega times (0,infty) ; , end{cases}]，其中$chi in mathbb{R}, m, gamma geq 1$和$a_0, a_1, a_2, alpha gt 0$。•当$chi gt 0$时，如果参数满足某些适当的假设，则上述系统的解是全局一致有界的。•当$chi gt 0$时，系统具有一个全局有界的经典解，假设$a_1 gt a_2 lvert Omega rvert$。这些结果表明，斥力机制在保证解的全局有界性中起着至关重要的作用。此外，通过构造能量泛函，导出了化学吸引或化学排斥系统全局有界解的大时间行为。
{"title":"On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term","authors":"Pan Zheng","doi":"10.4310/dpde.2024.v21.n1.a3","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n1.a3","url":null,"abstract":"and nonlocal growth term[begin{cases}u_t = Delta u - chi nabla cdot (u^m nabla v) + u Biggl( a_0 - a_1 u^alpha + a_2 displaystyle int_Omega u^sigma dx Biggr) & (x, t) in Omega times (0,infty) ; , 0=Delta - v + u^gamma , & (x, t) in Omega times (0,infty) ; , end{cases}]under homogeneous Neumann boundary conditions in a smoothly bounded domain $Omega subset mathbb{R}^n (n geq 1)$, where $chi in mathbb{R}, m, gamma geq 1$ and $a_0, a_1, a_2, alpha gt 0$. • When $chi gt 0$, the solution of the above system is global and uniformly bounded, if the parameters satisfy certain suitable assumptions. • When $chi gt 0$, the system possesses a globally bounded classical solution, provided that $a_1 gt a_2 lvert Omega rvert$. These results indicate that the repulsive mechanism plays a crucial role in ensuring the global boundedness of solutions. In addition, the paper derives the large time behavior of globally bounded solutions for the chemo-attractive or chemo-repulsive system by constructing energy functionals.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.4310/dpde.2024.v21.n1.a1
Lucas C. F. Ferreira, Daniel F. Machado
In this paper we study the inhomogeneous incompressible Euler equations in the whole space $mathbb{R}^n$ with $n geq 3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov–Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity. Comparing with previous works on Besov spaces, our results provide a larger initial data class for a well-defined flow. For that, we need to obtain suitable linear estimates for some conservation-law models in our setting such as transport equations and the linearized inhomogeneous Euler system.
{"title":"On the well-posedness in Besov–Herz spaces for the inhomogeneous incompressible Euler equations","authors":"Lucas C. F. Ferreira, Daniel F. Machado","doi":"10.4310/dpde.2024.v21.n1.a1","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n1.a1","url":null,"abstract":"In this paper we study the inhomogeneous incompressible Euler equations in the whole space $mathbb{R}^n$ with $n geq 3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov–Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity. Comparing with previous works on Besov spaces, our results provide a larger initial data class for a well-defined flow. For that, we need to obtain suitable linear estimates for some conservation-law models in our setting such as transport equations and the linearized inhomogeneous Euler system.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.4310/dpde.2024.v21.n1.a2
Yuxia Guo, Shaolong Peng
In this paper, we are concerned with the physically interesting static weighted Schrödinger equations involving logarithmic nonlinearities:[(-Delta)^s u = c_1 {lvert x rvert}^a u^{p_1} log (1 + u^{q_1}) + c_2 {lvert x rvert}^bBigl( dfrac{1}{{lvert : cdot : rvert}^sigma} Bigr) u^{p_2} quad textrm{,}]where $n geq 2, 0 lt s =: m + frac{alpha}{2} lt +infty , 0 lt alpha leq 2, 0 lt sigma lt n, c_1, c_2 geq 0$ with $c_1 + c_2 gt 0, 0 leq a, b lt+infty$. Here we point out the above equations involving higher-order or higher-order fractional Laplacians. We first derive the Liouville theorems (i.e., non-existence of nontrivial nonnegative solutions) in the subcritical-order cases (see Theorem 1.1) via the method of scaling spheres. Secondly, we obtain the Liouville-type results in critical and supercritical-order cases (see Theorem 1.2) by using some integral inequalities. As applications, we also derive Liouville-type results for the Schrödinger system involving logarithmic nonlinearities (see Theorem 1.4).
在本文中，我们关注物理上有趣的静态加权Schrödinger方程，涉及对数非线性:[(-Delta)^s u = c_1 {lvert x rvert}^a u^{p_1} log (1 + u^{q_1}) + c_2 {lvert x rvert}^bBigl( dfrac{1}{{lvert : cdot : rvert}^sigma} Bigr) u^{p_2} quad textrm{,}]其中$n geq 2, 0 lt s =: m + frac{alpha}{2} lt +infty , 0 lt alpha leq 2, 0 lt sigma lt n, c_1, c_2 geq 0$与$c_1 + c_2 gt 0, 0 leq a, b lt+infty$。这里我们指出上述方程涉及高阶或高阶分数拉普拉斯算子。我们首先通过标度球的方法推导了次临界阶情况下的Liouville定理(即非平凡非负解的不存在性)(见定理1.1)。其次，利用一些积分不等式得到临界阶和超临界阶情况下的liouville型结果(见定理1.2)。作为应用，我们还推导了涉及对数非线性的Schrödinger系统的liouville型结果(见定理1.4)。
{"title":"Liouville theorems for nonnegative solutions to weighted Schrödinger equations with logarithmic nonlinearities","authors":"Yuxia Guo, Shaolong Peng","doi":"10.4310/dpde.2024.v21.n1.a2","DOIUrl":"https://doi.org/10.4310/dpde.2024.v21.n1.a2","url":null,"abstract":"In this paper, we are concerned with the physically interesting static weighted Schrödinger equations involving logarithmic nonlinearities:[(-Delta)^s u = c_1 {lvert x rvert}^a u^{p_1} log (1 + u^{q_1}) + c_2 {lvert x rvert}^bBigl( dfrac{1}{{lvert : cdot : rvert}^sigma} Bigr) u^{p_2} quad textrm{,}]where $n geq 2, 0 lt s =: m + frac{alpha}{2} lt +infty , 0 lt alpha leq 2, 0 lt sigma lt n, c_1, c_2 geq 0$ with $c_1 + c_2 gt 0, 0 leq a, b lt+infty$. Here we point out the above equations involving higher-order or higher-order fractional Laplacians. We first derive the Liouville theorems (i.e., non-existence of nontrivial nonnegative solutions) in the subcritical-order cases (see Theorem 1.1) via the method of scaling spheres. Secondly, we obtain the Liouville-type results in critical and supercritical-order cases (see Theorem 1.2) by using some integral inequalities. As applications, we also derive Liouville-type results for the Schrödinger system involving logarithmic nonlinearities (see Theorem 1.4).","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/dpde.2023.v20.n3.a4
Min Gao, Jiao Xu
{"title":"Convergence to steady states of parabolic sine-Gordon","authors":"Min Gao, Jiao Xu","doi":"10.4310/dpde.2023.v20.n3.a4","DOIUrl":"https://doi.org/10.4310/dpde.2023.v20.n3.a4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/dpde.2023.v20.n3.a5
Mehdi Badra, Takéo Takahashi
{"title":"Analyticity of the semigroup corresponding to a strongly damped wave equation with a Ventcel boundary condition","authors":"Mehdi Badra, Takéo Takahashi","doi":"10.4310/dpde.2023.v20.n3.a5","DOIUrl":"https://doi.org/10.4310/dpde.2023.v20.n3.a5","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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