To study the dissipation property of a physical system one first considers infinitesimal waves for the analysis of weakly nonlinear phenomena. For some physical systems, the dissipation is partial and there is the appearance of sub-shocks in a strong traveling trajectory. The phenomenon of partial dissipation can occur for systems of hyperbolic balance laws and also for viscous conservation laws in continuum physics. We illustrate the phenomenon for a simple relaxation model and for the Navier-Stokes equations for compressible media. The admissibility criteria and the formation of sub-shocks are studied through the zero viscosity limit.
{"title":"Partial dissipation and sub-shock","authors":"Tai-Ping Liu","doi":"10.1090/qam/1657","DOIUrl":"https://doi.org/10.1090/qam/1657","url":null,"abstract":"To study the dissipation property of a physical system one first considers infinitesimal waves for the analysis of weakly nonlinear phenomena. For some physical systems, the dissipation is partial and there is the appearance of sub-shocks in a strong traveling trajectory. The phenomenon of partial dissipation can occur for systems of hyperbolic balance laws and also for viscous conservation laws in continuum physics. We illustrate the phenomenon for a simple relaxation model and for the Navier-Stokes equations for compressible media. The admissibility criteria and the formation of sub-shocks are studied through the zero viscosity limit.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46504883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preface for the special issue in honor of C. M. Dafermos","authors":"Govind Menon, M. Slemrod, A. Tzavaras","doi":"10.1090/qam/1653","DOIUrl":"https://doi.org/10.1090/qam/1653","url":null,"abstract":"","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45041682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into R 3 mathbb {R}^3 . The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a C 1 , 1 C^{1,1} isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in L 2 L^2 .
{"title":"Corrugated versus smooth uniqueness and stability of negatively curved isometric immersions","authors":"C. Christoforou","doi":"10.1090/qam/1663","DOIUrl":"https://doi.org/10.1090/qam/1663","url":null,"abstract":"We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 1\u0000 ,\u0000 1\u0000 \u0000 \u0000 C^{1,1}\u0000 \u0000\u0000 isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in \u0000\u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 L^2\u0000 \u0000\u0000.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48664482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λlambda-family equations, where λlambda is the power of nonlinear wave speed. The λlambda-family equations include Camassa-Holm equation (λ=1lambda =1) and Novikov equation (λ=2lambda =2) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1−12λ1- frac {1}{2lambda }. The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.
{"title":"Existence and regularity for global weak solutions to the 𝜆-family water wave equations","authors":"Geng Chen, Yannan Shen, Shihui Zhu","doi":"10.1090/qam/1660","DOIUrl":"https://doi.org/10.1090/qam/1660","url":null,"abstract":"<p>In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\u0000 <mml:semantics>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-family equations, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\u0000 <mml:semantics>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the power of nonlinear wave speed. The <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\u0000 <mml:semantics>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-family equations include Camassa-Holm equation (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda equals 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda =1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) and Novikov equation (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda equals 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda =2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 minus StartFraction 1 Over 2 lamda EndFraction\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">1- frac {1}{2lambda }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45323855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the a free energy. In this paper hyper-elastic Ricci flow is shown to possess three properties derived by G. Perelman for classical Ricci flow, specifically it is diffeomorphically equivalent to a gradient flow, unique smooth solutions exist locally in time, and the system possesses a non-decreasing energy function.
{"title":"Hyper-elastic Ricci flow: Gradient flow, local existence-uniqueness, and a Perelman energy functional","authors":"M. Slemrod","doi":"10.1090/qam/1643","DOIUrl":"https://doi.org/10.1090/qam/1643","url":null,"abstract":"The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the a free energy. In this paper hyper-elastic Ricci flow is shown to possess three properties derived by G. Perelman for classical Ricci flow, specifically it is diffeomorphically equivalent to a gradient flow, unique smooth solutions exist locally in time, and the system possesses a non-decreasing energy function.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45838855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Ancona, S. Bianchini, A. Bressan, R. Colombo, K. Nguyen
The paper discusses various regularity properties for solutions to a scalar, 1-dimensional conservation law with strictly convex flux and integrable source. In turn, these yield compactness estimates on the solution set. Similar properties are expected to hold for 2 × 2 2times 2 genuinely nonlinear systems.
{"title":"Examples and conjectures on the regularity of solutions to balance laws","authors":"F. Ancona, S. Bianchini, A. Bressan, R. Colombo, K. Nguyen","doi":"10.1090/qam/1647","DOIUrl":"https://doi.org/10.1090/qam/1647","url":null,"abstract":"The paper discusses various regularity properties for solutions to a scalar, 1-dimensional conservation law with strictly convex flux and integrable source. In turn, these yield compactness estimates on the solution set. Similar properties are expected to hold for \u0000\u0000 \u0000 \u0000 2\u0000 ×\u0000 2\u0000 \u0000 2times 2\u0000 \u0000\u0000 genuinely nonlinear systems.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43229034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space R n mathbb {R}^{n} . Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum.
研究了高维非齐次不可压缩对数超耗散Navier-Stokes方程的Cauchy问题。利用Littlewood-Paley技术和新思想，建立了在整个空间R n mathbb {R}^{n}上具有真空的全局强解的存在唯一性。此外，我们还得到了强解的指数时间衰减。我们的结果在初始数据上没有任何小，并且允许初始密度有真空。
{"title":"Global well-posedness and exponential decay for the inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation","authors":"Dehua Wang, Z. Ye","doi":"10.1090/qam/1644","DOIUrl":"https://doi.org/10.1090/qam/1644","url":null,"abstract":"We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 n\u0000 \u0000 \u0000 mathbb {R}^{n}\u0000 \u0000\u0000. Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42537033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. Buckmaster, Gonzalo Cao-Labora, Javier G'omez-Serrano
The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents γ > 1 gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for γ = 7 5 gamma =frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.
本文的目的是介绍Buckmaster, cho - labora和Gómez-Serrano[三维可压缩流体的光滑内爆解，Arxiv预印Arxiv:2208.09445, 2022]关于三维等熵可压缩Euler和Navier-Stokes方程“内爆奇异点”存在的最新结果。我们的工作建立在Merle, Raphaël, Rodnianski和Szeftel [Invent]的开创性工作之上。数学。227 (2022)，pp. 247-413;安。数学。(2) 196 (2022)， pp. 567-778;安。数学。(2) 196 (2022)， pp. 779-889]并证明了在欧拉情况下所有绝热指数γ >1 gamma >1的自相似分布的存在;以及证明了在Navier-Stokes情况下γ = 75 gamma = frac 75的渐近自相似爆破。重要的是，对于Navier-Stokes方程，其解被构造成密度有界，远离零，在无穷远处恒定，这是在这种情况下爆炸的第一个例子。为简单起见，我们将集中讨论可压缩欧拉方程。
{"title":"Smooth self-similar imploding profiles to 3D compressible Euler","authors":"T. Buckmaster, Gonzalo Cao-Labora, Javier G'omez-Serrano","doi":"10.1090/qam/1661","DOIUrl":"https://doi.org/10.1090/qam/1661","url":null,"abstract":"The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents \u0000\u0000 \u0000 \u0000 γ\u0000 >\u0000 1\u0000 \u0000 gamma >1\u0000 \u0000\u0000 in the case of Euler; as well as proving asymptotic self-similar blow-up for \u0000\u0000 \u0000 \u0000 γ\u0000 =\u0000 \u0000 7\u0000 5\u0000 \u0000 \u0000 gamma =frac 75\u0000 \u0000\u0000 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43104760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness. These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding special velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.
{"title":"Non-uniqueness in plane fluid flows","authors":"H. Gimperlein, M. Grinfeld, R. Knops, M. Slemrod","doi":"10.1090/qam/1670","DOIUrl":"https://doi.org/10.1090/qam/1670","url":null,"abstract":"Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness.\u0000\u0000These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding special velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48436621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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