Pub Date : 2020-10-19DOI: 10.1142/s0218202521500305
G. K. Duong, N. Kavallaris, H. Zaag
In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem begin{equation*} left{begin{array}{rcl} partial_t u &=& Delta u - u + displaystyle{frac{u^p}{ left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^gamma }}quadtext{in}quad Omega times (0,T), [0.2cm] frac{ partial u}{ partial nu} & = & 0 text{ on } Gamma = partial Omega times (0,T), u(0) & = & u_0, end{array} right. end{equation*} where $Omega$ is a bounded domain in $mathbb{R}^N$ with smooth boundary $partial Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. cite{KSN17, NKMI2018}. Under the Turing type condition $$ frac{r}{p-1} < frac{N}{2}, gamma r ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $Omega,$ i.e. $$ u(x_0, t) sim (theta^*)^{-frac{1}{p-1}} left[kappa (T-t)^{-frac{1}{p-1}} right], $$ where $$ theta^* := lim_{t to T} left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^{- gamma} text{ and } kappa = (p-1)^{-frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) sim ( theta^* )^{-frac{1}{p-1}} left[ frac{(p-1)^2}{8p} frac{|x-x_0|^2}{ |ln|x-x_0||} right]^{ -frac{1}{p-1}} text{ as } x to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in cite{MZnon97} and cite{DZM3AS19}.
在本文中，我们对以下非局部问题begin{equation*} left{begin{array}{rcl} partial_t u &=& Delta u - u + displaystyle{frac{u^p}{ left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^gamma }}quadtext{in}quad Omega times (0,T), [0.2cm] frac{ partial u}{ partial nu} & = & 0 text{ on } Gamma = partial Omega times (0,T), u(0) & = & u_0, end{array} right. end{equation*}的解的扩散引起的爆炸行为进行了深入的研究，其中$Omega$是$mathbb{R}^N$中具有光滑边界的有界区域$partial Omega;$，该问题被导出为奇异Gierer-Meinhardt系统的阴影极限，参见cite{KSN17, NKMI2018}。在图灵型条件$$ frac{r}{p-1} < frac{N}{2}, gamma r ne p-1, $$下，我们构造了一个在有限时间内只在$Omega,$的内部点$x_0$爆炸的解，即$$ u(x_0, t) sim (theta^*)^{-frac{1}{p-1}} left[kappa (T-t)^{-frac{1}{p-1}} right], $$，其中$$ theta^* := lim_{t to T} left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^{- gamma} text{ and } kappa = (p-1)^{-frac{1}{p-1}}. $$更准确地说，我们还给出了爆炸点$$ u(x,T) sim ( theta^* )^{-frac{1}{p-1}} left[ frac{(p-1)^2}{8p} frac{|x-x_0|^2}{ |ln|x-x_0||} right]^{ -frac{1}{p-1}} text{ as } x to 0, $$的最终渐近轮廓的描述，从而揭示了由于驱动扩散不稳定而在这种情况下发生的图灵模式的形式。构建上述爆破溶液的应用技术主要依靠cite{MZnon97}和cite{DZM3AS19}中开发的方法。
{"title":"Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system","authors":"G. K. Duong, N. Kavallaris, H. Zaag","doi":"10.1142/s0218202521500305","DOIUrl":"https://doi.org/10.1142/s0218202521500305","url":null,"abstract":"In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem begin{equation*} left{begin{array}{rcl} partial_t u &=& Delta u - u + displaystyle{frac{u^p}{ left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^gamma }}quadtext{in}quad Omega times (0,T), [0.2cm] frac{ partial u}{ partial nu} & = & 0 text{ on } Gamma = partial Omega times (0,T), u(0) & = & u_0, end{array} right. end{equation*} where $Omega$ is a bounded domain in $mathbb{R}^N$ with smooth boundary $partial Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. cite{KSN17, NKMI2018}. Under the Turing type condition $$ frac{r}{p-1} < frac{N}{2}, gamma r ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $Omega,$ i.e. $$ u(x_0, t) sim (theta^*)^{-frac{1}{p-1}} left[kappa (T-t)^{-frac{1}{p-1}} right], $$ where $$ theta^* := lim_{t to T} left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^{- gamma} text{ and } kappa = (p-1)^{-frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) sim ( theta^* )^{-frac{1}{p-1}} left[ frac{(p-1)^2}{8p} frac{|x-x_0|^2}{ |ln|x-x_0||} right]^{ -frac{1}{p-1}} text{ as } x to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in cite{MZnon97} and cite{DZM3AS19}.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91130707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.
{"title":"Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations","authors":"Ludovic Godard-Cadillac","doi":"10.5802/CRMATH.159","DOIUrl":"https://doi.org/10.5802/CRMATH.159","url":null,"abstract":"In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89586794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peral/Miyachi’s celebrated theorem on fixed time $L^p$ estimates with loss of derivatives for the wave equation states that the operator $(I-Delta)^{-frac{alpha}{2}}exp(isqrt{-Delta})$ is bounded on $L^p(mathbb{R}^d)$ if and only if $alphage s_p:=(d-1)left|frac{1}{p}-frac{1}{2}right|$. We extend this result tooperators of the form $L=−displaystylesum_{j=1}^d a_jpartial_j a_jpartial_j$, for functions $xmapsto a_i(x_i)$ that are bounded above and below, but merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $dge2$. Our proof is based on an approach to the boundedness of Fourier integral operators recently developed by Hassell, Rozendaal, and the second author. We construct a scale of adapted Hardy spaces on which $exp(isqrt{L})$ is bounded by lifting $L^p$ functions to the tent space $T^{p,2}(mathbb{R}^d)$, using a wave packet transform adapted to the Lipschitz metric induced by $A$. The result then follows from Sobolev embedding properties of these spaces.
{"title":"$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients","authors":"D. Frey, Pierre Portal","doi":"10.5445/IR/1000124653","DOIUrl":"https://doi.org/10.5445/IR/1000124653","url":null,"abstract":"Peral/Miyachi’s celebrated theorem on fixed time $L^p$ estimates with loss of derivatives for the wave equation states that the operator $(I-Delta)^{-frac{alpha}{2}}exp(isqrt{-Delta})$ is bounded on $L^p(mathbb{R}^d)$ if and only if $alphage s_p:=(d-1)left|frac{1}{p}-frac{1}{2}right|$. We extend this result tooperators of the form $L=−displaystylesum_{j=1}^d a_jpartial_j a_jpartial_j$, for functions $xmapsto a_i(x_i)$ that are bounded above and below, but merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $dge2$. Our proof is based on an approach to the boundedness of Fourier integral operators recently developed by Hassell, Rozendaal, and the second author. We construct a scale of adapted Hardy spaces on which $exp(isqrt{L})$ is bounded by lifting $L^p$ functions to the tent space $T^{p,2}(mathbb{R}^d)$, using a wave packet transform adapted to the Lipschitz metric induced by $A$. The result then follows from Sobolev embedding properties of these spaces.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"114 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79032355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain space-time Holder regularity estimates for solutions of first-and second-order Hamilton-Jacobi equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.
{"title":"Hölder regularity of Hamilton-Jacobi equations with stochastic forcing","authors":"P. Cardaliaguet, B. Seeger","doi":"10.1090/TRAN/8435","DOIUrl":"https://doi.org/10.1090/TRAN/8435","url":null,"abstract":"We obtain space-time Holder regularity estimates for solutions of first-and second-order Hamilton-Jacobi equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88970446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a previous work on the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations, we treated in detail the situation when a certain potential is the characteristic function of a strictly convex set with real-analytic boundary. We here extend the results to the case of sets with smooth boundary, by using almost holomorphic functions.
{"title":"Large $|k|$ behavior of d-bar problems for domains with a smooth boundary","authors":"C. Klein, J. Sjostrand, N. Stoilov","doi":"10.4171/ecr/18-1/15","DOIUrl":"https://doi.org/10.4171/ecr/18-1/15","url":null,"abstract":"In a previous work on the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations, we treated in detail the situation when a certain potential is the characteristic function of a strictly convex set with real-analytic boundary. We here extend the results to the case of sets with smooth boundary, by using almost holomorphic functions.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84788299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-08DOI: 10.22541/au.160395665.59674549/v1
M. Hamouda, M. Hamza
We are interested in this article in studying the damped wave equation with localized initial data, in the textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-Delta u+frac{mu}{1+t}u_t+frac{nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty), $$ with small initial data. Under some assumptions on the mass and damping coefficients, $nu$ and $mu>0$, respectively, we show that blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained in cite{Our2} in the case of a mass-free wave equation, it i.e. $(E)$ with $nu=0$. Furthermore, using in part the computations done for $(E)$, we enhance the result in cite{Palmieri} on the Glassey conjecture for the solution of $(E)$ with omitting the nonlinear term $|u|^q$. Indeed, the blow-up region is extended from $p in (1, p_G(N+sigma)]$, where $sigma$ is given by (1.12) below, to $p in (1, p_G(N+mu)]$ yielding, hence, a better estimate of the lifespan when $(mu-1)^2-4nu^2<1$. Otherwise, the two results coincide. Finally, we may conclude that the mass term {it has no influence} on the dynamics of $(E)$ (resp. $(E)$ without the nonlinear term $|u|^q$), and the conjecture we made in cite{Our2} on the threshold between the blow-up and the global existence regions obtained holds true here.
本文主要研究具有局部初始数据的阻尼波动方程，在具有质量项和两种组合非线性的textit{尺度不变情况下}。更准确地说，我们考虑以下等式:$$ (E) {1cm} u_{tt}-Delta u+frac{mu}{1+t}u_t+frac{nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty), $$初始数据较小。在质量和阻尼系数分别为$nu$和$mu>0$的一些假设下，我们表明，在无质量波动方程的情况下，$(E)$的解的爆炸区域和寿命界与cite{Our2}中得到的相同，即$(E)$与$nu=0$。此外，利用对$(E)$所做的部分计算，我们增强了cite{Palmieri}中关于$(E)$解的Glassey猜想的结果，省略了非线性项$|u|^q$。实际上，爆炸区域从$p in (1, p_G(N+sigma)]$延伸到$p in (1, p_G(N+mu)]$，其中$sigma$由下面的(1.12)给出，因此，对$(mu-1)^2-4nu^2<1$时的寿命有一个更好的估计。否则，两个结果是一致的。最后，我们可以得出结论，质量{it项对}$(E)$的动力学没有影响。$(E)$没有非线性项$|u|^q$)，并且我们在cite{Our2}中所做的关于爆炸和得到的整体存在区域之间的阈值的猜想在这里成立。
{"title":"A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities","authors":"M. Hamouda, M. Hamza","doi":"10.22541/au.160395665.59674549/v1","DOIUrl":"https://doi.org/10.22541/au.160395665.59674549/v1","url":null,"abstract":"We are interested in this article in studying the damped wave equation with localized initial data, in the textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-Delta u+frac{mu}{1+t}u_t+frac{nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty), $$ with small initial data. Under some assumptions on the mass and damping coefficients, $nu$ and $mu>0$, respectively, we show that blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained in cite{Our2} in the case of a mass-free wave equation, it i.e. $(E)$ with $nu=0$. \u0000Furthermore, using in part the computations done for $(E)$, we enhance the result in cite{Palmieri} on the Glassey conjecture for the solution of $(E)$ with omitting the nonlinear term $|u|^q$. Indeed, the blow-up region is extended from $p in (1, p_G(N+sigma)]$, where $sigma$ is given by (1.12) below, to $p in (1, p_G(N+mu)]$ yielding, hence, a better estimate of the lifespan when $(mu-1)^2-4nu^2<1$. Otherwise, the two results coincide. Finally, we may conclude that the mass term {it has no influence} on the dynamics of $(E)$ (resp. $(E)$ without the nonlinear term $|u|^q$), and the conjecture we made in cite{Our2} on the threshold between the blow-up and the global existence regions obtained holds true here.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79092784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Antonio J. Fern'andez, L. Jeanjean, Rainer Mandel, M. Mariş
We study the standing waves for a fourth-order Schrodinger equation with mixed dispersion that minimize the associated energy when the $L^2$-norm (the $textit{mass}$) is kept fixed. We need some non-homogeneous Gagliardo−Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the $textit{mass-subcritical}$ and $textit{mass-critical}$ cases. In the $textit{mass super-critical}$ case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $μ_0in (0,+infty)$, our results on "best" local minimizers are also optimal.
{"title":"Some non-homogeneous Gagliardo–Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation","authors":"Antonio J. Fern'andez, L. Jeanjean, Rainer Mandel, M. Mariş","doi":"10.5445/IR/1000124276","DOIUrl":"https://doi.org/10.5445/IR/1000124276","url":null,"abstract":"We study the standing waves for a fourth-order Schrodinger equation with mixed dispersion that minimize the associated energy when the $L^2$-norm (the $textit{mass}$) is kept fixed. We need some non-homogeneous Gagliardo−Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the $textit{mass-subcritical}$ and $textit{mass-critical}$ cases. In the $textit{mass super-critical}$ case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $μ_0in (0,+infty)$, our results on \"best\" local minimizers are also optimal.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88258631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension $2$ we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.
{"title":"(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property","authors":"S. Dipierro, F. Onoue, E. Valdinoci","doi":"10.1090/proc/15796","DOIUrl":"https://doi.org/10.1090/proc/15796","url":null,"abstract":"We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. \u0000Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension $2$ we provide a quantitative bound on the stickiness property exhibited by the minimizers. \u0000Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85013044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite a non-vanishing density by considering a negatively unbounded entropy and we use a novel weighted energy method whereby the exponential of the entropy will act as a changing weight to handle the degeneracy of the vacuum boundary. Spherical symmetry introduces a coordinate singularity near the origin for which we adapt a method developed for the Euler-Poisson system by Guo, Hadžic and Jang to our problem.
{"title":"The vacuum boundary problem for the spherically symmetric compressible Euler equations with positive density and unbounded entropy","authors":"C. Rickard","doi":"10.1063/5.0037656","DOIUrl":"https://doi.org/10.1063/5.0037656","url":null,"abstract":"Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite a non-vanishing density by considering a negatively unbounded entropy and we use a novel weighted energy method whereby the exponential of the entropy will act as a changing weight to handle the degeneracy of the vacuum boundary. Spherical symmetry introduces a coordinate singularity near the origin for which we adapt a method developed for the Euler-Poisson system by Guo, Hadžic and Jang to our problem.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79497671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：