Communications on Pure & Applied Analysis最新文献

{"title":"Energy considerations for nonlinear equatorial water waves","authors":"D. Henry","doi":"10.3934/cpaa.2022057","DOIUrl":"https://doi.org/10.3934/cpaa.2022057","url":null,"abstract":"In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127132044","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}

{"title":"Liouville type theorem for Hartree-Fock Equation on half space","authors":"Xiaomei Chen, Xiaohui Yu","doi":"10.3934/cpaa.2022050","DOIUrl":"https://doi.org/10.3934/cpaa.2022050","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{align*} begin{cases} - Delta {u_i}(y) = sumlimits_{j = 1}^n {{int _{partial mathbb{R}_ + ^N}}} frac{{{u_j}(bar x, 0){F_1}({u_j}(bar x, 0))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dbar x{f_2}({u_i}(y)) qquad qquad qquad + sumlimits_{j = 1}^n {{int _{partial mathbb{R}_ + ^N}}} frac{{{u_j}(bar x, 0){F_2}({u_i}(bar x, 0))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dbar x{f_1}({u_j}(y)), y in mathbb{R}_ + ^N, hfill frac{{partial {u_i}}} {{partial nu }}(bar x, 0) = sumlimits_{j = 1}^n {{int _{ mathbb{R}_ + ^N}}} frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dy{g_2}({u_i}(bar x, 0)) qquad qquad qquad + sumlimits_{j = 1}^n {{int _{ mathbb{R}_ + ^N}}} frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dy{g_1}({u_j}(bar x, 0)), quad quad(bar x, 0) in partial mathbb{R}_ + ^N, end{cases} end{align*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}_+^N = {xin{mathbb{R}^N}: x_N > 0}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $end{document}</tex-math></inline-formula> are some nonlinear functions. Under some assumptions on the nonlinear functions <inline-formula><tex-math id=\"M2\">begin{document}$ F, G, f, g $end{document}</tex-math></inline-formula>, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"282 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129989422","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}

{"title":"On three-dimensional free surface water flows with constant vorticity","authors":"Calin Iulian Martin","doi":"10.3934/cpaa.2022053","DOIUrl":"https://doi.org/10.3934/cpaa.2022053","url":null,"abstract":"We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125207926","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}

{"title":"Exact solution and instability for geophysical edge waves","authors":"Fahe Miao, Michal Feckan, Jinrong Wang","doi":"10.3934/cpaa.2022067","DOIUrl":"https://doi.org/10.3934/cpaa.2022067","url":null,"abstract":"<p style='text-indent:20px;'>We present an exact solution to the nonlinear governing equations in the <inline-formula><tex-math id=\"M1\">begin{document}$ beta $end{document}</tex-math></inline-formula>-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133890960","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}

{"title":"Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains","authors":"Shu Wang, Mengmeng Si, Rong Yang","doi":"10.3934/cpaa.2022034","DOIUrl":"https://doi.org/10.3934/cpaa.2022034","url":null,"abstract":"In this paper, we study the asymptotic behavior of the non-autono-mous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132455368","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}

{"title":"On the spherical geopotential approximation for Saturn","authors":"Susanna V. Haziot","doi":"10.3934/cpaa.2022035","DOIUrl":"https://doi.org/10.3934/cpaa.2022035","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we show by means of a diffeomorphism that when approximating the planet Saturn by a sphere, the errors associated with the spherical geopotential approximation are so significant that this approach is rendered unsuitable for any rigorous mathematical analysis.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130688341","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}

{"title":"Singular quasilinear critical Schrödinger equations in $ mathbb {R}^N $","authors":"Laura Baldelli, Roberta Filippucci","doi":"10.3934/cpaa.2022060","DOIUrl":"https://doi.org/10.3934/cpaa.2022060","url":null,"abstract":"<p style='text-indent:20px;'>We prove multiplicity results for solutions, both with positive and negative energy, for a class of singular quasilinear Schrödinger equations in the entire <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb {R}^N $end{document}</tex-math></inline-formula> involving a critical term, nontrivial weights and positive parameters <inline-formula><tex-math id=\"M3\">begin{document}$ lambda $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M4\">begin{document}$ beta $end{document}</tex-math></inline-formula>, covering several physical models, coming from plasma physics as well as high-power ultra short laser in matter. Also the symmetric setting is investigated. Our proofs relay on variational tools, including concentration compactness principles because of the delicate situation of the double lack of compactness. In addition, a necessary reformulation of the original problem in a suitable variational setting, produces a singular function, delicate to be managed.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124999990","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}

{"title":"Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials","authors":"Xiaoming An, Xian Yang","doi":"10.3934/cpaa.2022038","DOIUrl":"https://doi.org/10.3934/cpaa.2022038","url":null,"abstract":"<p style='text-indent:20px;'>This paper deals with the following fractional magnetic Schrödinger equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ varepsilon^{2s}(-Delta)^s_{A/varepsilon} u +V(x)u = |u|^{p-2}u, xin{mathbb R}^N, $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ varepsilon>0 $end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id=\"M2\">begin{document}$ sin(0,1) $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M3\">begin{document}$ Ngeq3 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M4\">begin{document}$ 2+2s/(N-2s)<p<2_s^*: = 2N/(N-2s) $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M5\">begin{document}$ Ain C^{0,alpha}({mathbb R}^N,{mathbb R}^N) $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M6\">begin{document}$ alphain(0,1] $end{document}</tex-math></inline-formula> is a magnetic field, <inline-formula><tex-math id=\"M7\">begin{document}$ V:{mathbb R}^Nto{mathbb R} $end{document}</tex-math></inline-formula> is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of <inline-formula><tex-math id=\"M8\">begin{document}$ V $end{document}</tex-math></inline-formula> as <inline-formula><tex-math id=\"M9\">begin{document}$ varepsilonto 0 $end{document}</tex-math></inline-formula>. There is no restriction on the decay rates of <inline-formula><tex-math id=\"M10\">begin{document}$ V $end{document}</tex-math></inline-formula>. Especially, <inline-formula><tex-math id=\"M11\">begin{document}$ V $end{document}</tex-math></inline-formula> can be compactly supported. The appearance of <inline-formula><tex-math id=\"M12\">begin{document}$ A $end{document}</tex-math></inline-formula> and the nonlocal of <inline-formula><tex-math id=\"M13\">begin{document}$ (-Delta)^s $end{document}</tex-math></inline-formula> makes the proof more difficult than that in [<xref ref-type=\"bibr\" rid=\"b7\">7</xref>], which considered the case <inline-formula><tex-math id=\"M14\">begin{document}$ Aequiv 0 $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116828628","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}

{"title":"Analysis of one-sided 1-D fractional diffusion operator","authors":"Yulong Li, A. Telyakovskiy, E. Celik","doi":"10.3934/cpaa.2022039","DOIUrl":"https://doi.org/10.3934/cpaa.2022039","url":null,"abstract":"This work establishes the parallel between the properties of classic elliptic PDEs and the one-sided 1-D fractional diffusion equation, that includes the characterization of fractional Sobolev spaces in terms of fractional Riemann-Liouville (R-L) derivatives, variational formulation, maximum principle, Hopf's Lemma, spectral analysis, and theory on the principal eigenvalue and its characterization, etc. As an application, the developed results provide a novel perspective to study the distribution of complex roots of a class of Mittag-Leffler functions and, furthermore, prove the existence of real roots.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129203244","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}

{"title":"Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems","authors":"Xueying Chen, Guanfeng Li, Sijia Bao","doi":"10.3934/cpaa.2022045","DOIUrl":"https://doi.org/10.3934/cpaa.2022045","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we focus on a class of general pseudo-relativistic systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} begin{cases} begin{aligned} &(-Delta+m^2)^su(x) = f(u(x), v(x)), &(-Delta+m^2)^tv(x) = g(u(x), v(x)), end{aligned} end{cases} end{equation*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ m in (0, +infty) $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M2\">begin{document}$ s, t in (0,1) $end{document}</tex-math></inline-formula>. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128263674","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}