We prove uniform H"older estimates in a class of singularly perturbed competition-diffusion elliptic systems, with the particular feature that the interactions between the components occur three by three (ternary interactions). These systems are associated to the minimization of Gross-Pitaevski energies modeling ternary mixture of ultracold gases and other multicomponent liquids and gases. We address the question whether this regularity holds uniformly throughout the approximation process up to the limiting profiles, answering positively. A very relevant feature of limiting profiles in this process is that they are only partially segregated, giving rise to new phenomena of geometric pattern formation and optimal regularity.
{"title":"On some singularly perturbed elliptic systems modeling partial segregation, Part 1: uniform Hölder estimates and basic properties of the limits","authors":"Nicola Soave, Susanna Terracini","doi":"arxiv-2409.11976","DOIUrl":"https://doi.org/arxiv-2409.11976","url":null,"abstract":"We prove uniform H\"older estimates in a class of singularly perturbed\u0000competition-diffusion elliptic systems, with the particular feature that the\u0000interactions between the components occur three by three (ternary\u0000interactions). These systems are associated to the minimization of\u0000Gross-Pitaevski energies modeling ternary mixture of ultracold gases and other\u0000multicomponent liquids and gases. We address the question whether this\u0000regularity holds uniformly throughout the approximation process up to the\u0000limiting profiles, answering positively. A very relevant feature of limiting\u0000profiles in this process is that they are only partially segregated, giving\u0000rise to new phenomena of geometric pattern formation and optimal regularity.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260614","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 a Kelvin-Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows.
{"title":"Nonlinear relations of viscous stress and strain rate in nonlinear Viscoelasticity","authors":"Lennart Machill","doi":"arxiv-2409.11882","DOIUrl":"https://doi.org/arxiv-2409.11882","url":null,"abstract":"We consider a Kelvin-Voigt model for viscoelastic second-grade materials,\u0000where the elastic and the viscous stress tensor both satisfy frame\u0000indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of\u0000weak solutions is shown by means of a frame-indifferent time-discretization\u0000scheme. Further, the result includes viscous stress tensors which can be\u0000calculated by nonquadratic polynomial densities. Afterwards, we investigate the\u0000long-time behavior of solutions in the case of small external loading and\u0000initial data. Our main tool is the abstract theory of metric gradient flows.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260613","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}
A diffuse-interface model that describes the dynamics of nonhomogeneous incompressible two-phase viscous flows is investigated in a bounded smooth domain in ${mathbb R}^3.$ The dynamics of the state variables is described by the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system. We first give a blow-up criterion of local strong solution to the initial-boundary-value problem for the case of initial density away from zero. After establishing some key a priori with the help of the Landau Potential, we obtain the global existence and decay-in-time of strong solution, provided that the initial date $|nabla u_0|_{L^{2}(Omega)}+|nabla mu_0|_{L^{2}(Omega)}+rho_0$ is suitably small.
{"title":"Global well-posedness of the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with Landau Potential","authors":"Nie Rui, Fang Li, Guo Zhenhua","doi":"arxiv-2409.11775","DOIUrl":"https://doi.org/arxiv-2409.11775","url":null,"abstract":"A diffuse-interface model that describes the dynamics of nonhomogeneous\u0000incompressible two-phase viscous flows is investigated in a bounded smooth\u0000domain in ${mathbb R}^3.$ The dynamics of the state variables is described by\u0000the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system. We first\u0000give a blow-up criterion of local strong solution to the initial-boundary-value\u0000problem for the case of initial density away from zero. After establishing some\u0000key a priori with the help of the Landau Potential, we obtain the global\u0000existence and decay-in-time of strong solution, provided that the initial date\u0000$|nabla u_0|_{L^{2}(Omega)}+|nabla mu_0|_{L^{2}(Omega)}+rho_0$ is\u0000suitably small.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260616","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 this article we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory, vanishing order of eigenfunctions, and unique continuation. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, orientable Riemannian manifold as an emph{imbedded metric graph} and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the $k$-th eigenfunction and the sum of vanishing orders at critical points in terms of $k$ and the genus of the manifold.
{"title":"Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds","authors":"Matthias Hofmann, Matthias Täufer","doi":"arxiv-2409.11800","DOIUrl":"https://doi.org/arxiv-2409.11800","url":null,"abstract":"In this article we illustrate and draw connections between the geometry of\u0000zero sets of eigenfunctions, graph theory, vanishing order of eigenfunctions,\u0000and unique continuation. We identify the nodal set of an eigenfunction of the\u0000Laplacian (with smooth potential) on a compact, orientable Riemannian manifold\u0000as an emph{imbedded metric graph} and then use tools from elementary graph\u0000theory in order to estimate the number of critical points in the nodal set of\u0000the $k$-th eigenfunction and the sum of vanishing orders at critical points in\u0000terms of $k$ and the genus of the manifold.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260617","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}
Linus Behn, Lars Diening, Jihoon Ok, Julian Rolfes
We introduce fractional weighted Sobolev spaces with degenerate weights. For these spaces we provide embeddings and Poincar'e inequalities. When the order of fractional differentiability goes to $0$ or $1$, we recover the weighted Lebesgue and Sobolev spaces with Muckenhoupt weights, respectively. Moreover, we prove interior H"older continuity and Harnack inequalities for solutions to the corresponding weighted nonlocal integro-differential equations. This naturally extends a classical result by Fabes, Kenig, and Serapioni to the nonlinear, nonlocal setting.
{"title":"Nonlocal equations with degenerate weights","authors":"Linus Behn, Lars Diening, Jihoon Ok, Julian Rolfes","doi":"arxiv-2409.11829","DOIUrl":"https://doi.org/arxiv-2409.11829","url":null,"abstract":"We introduce fractional weighted Sobolev spaces with degenerate weights. For\u0000these spaces we provide embeddings and Poincar'e inequalities. When the order\u0000of fractional differentiability goes to $0$ or $1$, we recover the weighted\u0000Lebesgue and Sobolev spaces with Muckenhoupt weights, respectively. Moreover,\u0000we prove interior H\"older continuity and Harnack inequalities for solutions to\u0000the corresponding weighted nonlocal integro-differential equations. This\u0000naturally extends a classical result by Fabes, Kenig, and Serapioni to the\u0000nonlinear, nonlocal setting.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260615","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}
The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system begin{align*} left{ begin{array}{r@{,}l@{quad}l@{,}c} &u_{t}=nablacdotbig(u^{m-1}vnabla vbig)-nablacdotbig(f(u)vnabla vbig)+ell uv, &v_{t}=Delta v-uv, end{array}right.%} end{align*} under no-flux boundary conditions in a smoothly bounded convex domain $Omegasubset R^2$, where the nonnegative function $fin C^1([0,infty))$ is asked to satisfy $f(s)le C_fs^{al}$ with $al, C_f>0$ for all $sge 1$. The global existence of weak solutions or classical solutions to the above system has been established in both one- and two-dimensional bounded convex domains in previous works. However, the results concerning the large time behavior are still constrained to one dimension due to the lack of a Harnack-type inequality in the two-dimensional case. In this note, we complement this result by using the Moser iteration technique and building a new Harnack-type inequality.
{"title":"The asymptotic behavior of solutions to a doubly degenerate chemotaxis-consumption system in two-dimensional setting","authors":"Duan Wu","doi":"arxiv-2409.12083","DOIUrl":"https://doi.org/arxiv-2409.12083","url":null,"abstract":"The present work proceeds to consider the convergence of the solutions to the\u0000following doubly degenerate chemotaxis-consumption system begin{align*}\u0000left{ begin{array}{r@{,}l@{quad}l@{,}c}\u0000&u_{t}=nablacdotbig(u^{m-1}vnabla vbig)-nablacdotbig(f(u)vnabla\u0000vbig)+ell uv, &v_{t}=Delta v-uv, end{array}right.%} end{align*} under\u0000no-flux boundary conditions in a smoothly bounded convex domain $Omegasubset\u0000R^2$, where the nonnegative function $fin C^1([0,infty))$ is asked to\u0000satisfy $f(s)le C_fs^{al}$ with $al, C_f>0$ for all $sge 1$. The global existence of weak solutions or classical solutions to the above\u0000system has been established in both one- and two-dimensional bounded convex\u0000domains in previous works. However, the results concerning the large time\u0000behavior are still constrained to one dimension due to the lack of a\u0000Harnack-type inequality in the two-dimensional case. In this note, we\u0000complement this result by using the Moser iteration technique and building a\u0000new Harnack-type inequality.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260612","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 prove an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,gamma}$ obstacle. In particular, we obtain the rate of convergence of the blow-up sequences at points of odd frequencies and the regularity of the strata of the corresponding contact set. We also recover the frequency gap for odd frequencies obtained by Savin and Yu.
{"title":"An epiperimetric inequality for odd frequencies in the thin obstacle problem","authors":"Matteo Carducci, Bozhidar Velichkov","doi":"arxiv-2409.12110","DOIUrl":"https://doi.org/arxiv-2409.12110","url":null,"abstract":"We prove an epiperimetric inequality for the thin obstacle Weiss' energy with\u0000odd frequencies and we apply it to solutions to the thin obstacle problem with\u0000general $C^{k,gamma}$ obstacle. In particular, we obtain the rate of\u0000convergence of the blow-up sequences at points of odd frequencies and the\u0000regularity of the strata of the corresponding contact set. We also recover the\u0000frequency gap for odd frequencies obtained by Savin and Yu.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260611","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}
The local and global well-posedness for the one dimensional fourth-order nonlinear Schr"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $sgeq frac12$ and $2leq q
针对 $sgeq frac12$ 和 $2leq q
{"title":"Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation","authors":"Mingjuan Chen, Yufeng Lu, Yaqing Wang","doi":"arxiv-2409.11002","DOIUrl":"https://doi.org/arxiv-2409.11002","url":null,"abstract":"The local and global well-posedness for the one dimensional fourth-order\u0000nonlinear Schr\"odinger equation are established in the modulation space\u0000$M^{s}_{2,q}$ for $sgeq frac12$ and $2leq q <infty$. The local result is\u0000based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key\u0000ingredient to obtain the global well-posedness is that we achieve a-priori\u0000estimates of the solution in modulation spaces by utilizing the power series\u0000expansion of the perturbation determinant introduced by Killip-Visan-Zhang for\u0000completely integrable PDEs.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260665","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}
Adina Ciomaga, Tri Minh Le, Olivier Ley, Erwin Topp
We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integro-differential operators in L'evy form, with general measures: $x$-dependent, possibly degenerate and without any restriction on the order. The measures must satisfy a combined Wasserstein/Total Variation-continuity assumption, which is one of the weakest conditions used in the context of viscosity approach for this type of integro-differential PDEs. The proof relies on a regularizing effect due to the gradient growth. We present several examples of applications to PDEs with different types of nonlocal operators (measures with density, operators of variable order, L'evy-It^o operators).
{"title":"Comparison principle for general nonlocal Hamilton-Jacobi equations with superlinear gradient","authors":"Adina Ciomaga, Tri Minh Le, Olivier Ley, Erwin Topp","doi":"arxiv-2409.11124","DOIUrl":"https://doi.org/arxiv-2409.11124","url":null,"abstract":"We obtain the comparison principle for discontinuous viscosity sub- and\u0000supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and\u0000coercive gradient terms. The nonlocal terms are integro-differential operators\u0000in L'evy form, with general measures: $x$-dependent, possibly degenerate and\u0000without any restriction on the order. The measures must satisfy a combined\u0000Wasserstein/Total Variation-continuity assumption, which is one of the weakest\u0000conditions used in the context of viscosity approach for this type of\u0000integro-differential PDEs. The proof relies on a regularizing effect due to the\u0000gradient growth. We present several examples of applications to PDEs with\u0000different types of nonlocal operators (measures with density, operators of\u0000variable order, L'evy-It^o operators).","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260662","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 study the two-dimensional MHD boundary layer equations. For small perturbation around a tangential background magnetic field, we obtain the global-in-time existence and uniqueness of solutions in Sobolev spaces. The proof relies on the novel combination of the well-explored cancellation mechanism and the idea of linearly-good unknowns, and in fact we use the former idea to deal with the top tangential derivatives and the latter one admitting fast decay rate to control lower-order derivatives.
{"title":"Global well-posedness of the MHD boundary layer equation in the Sobolev Space","authors":"Wei-Xi Li, Zhan Xu, Anita Yang","doi":"arxiv-2409.11009","DOIUrl":"https://doi.org/arxiv-2409.11009","url":null,"abstract":"We study the two-dimensional MHD boundary layer equations. For small\u0000perturbation around a tangential background magnetic field, we obtain the\u0000global-in-time existence and uniqueness of solutions in Sobolev spaces. The\u0000proof relies on the novel combination of the well-explored cancellation\u0000mechanism and the idea of linearly-good unknowns, and in fact we use the former\u0000idea to deal with the top tangential derivatives and the latter one admitting\u0000fast decay rate to control lower-order derivatives.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260664","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}
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