Journal of Evolution Equations最新文献

{"title":"Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions","authors":"Maykel Belluzi","doi":"10.1007/s00028-024-00961-y","DOIUrl":"https://doi.org/10.1007/s00028-024-00961-y","url":null,"abstract":"<p>In this work, we consider parabolic equations of the form </p><span>$$begin{aligned} (u_{varepsilon })_t +A_{varepsilon }(t)u_{{varepsilon }} = F_{varepsilon } (t,u_{{varepsilon } }), end{aligned}$$</span><p>where <span>(varepsilon )</span> is a parameter in <span>([0,varepsilon _0))</span>, and <span>({A_{varepsilon }(t), tin {mathbb {R}}})</span> is a family of uniformly sectorial operators. As <span>(varepsilon rightarrow 0^{+})</span>, we assume that the equation converges to </p><span>$$begin{aligned} u_t +A_{0}(t)u_{} = F_{0} (t,u_{}). end{aligned}$$</span><p>The time-dependence found on the linear operators <span>(A_{varepsilon }(t))</span> implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family <span>(A_{varepsilon }(t))</span> and on its convergence to <span>(A_0(t))</span> when <span>(varepsilon rightarrow 0^{+})</span>, we obtain a Trotter-Kato type Approximation Theorem for the linear process <span>(U_{varepsilon }(t,tau ))</span> associated with <span>(A_{varepsilon }(t))</span>, estimating its convergence to the linear process <span>(U_0(t,tau ))</span> associated with <span>(A_0(t))</span>. Through the variation of constants formula and assuming that <span>(F_{varepsilon })</span> converges to <span>(F_0)</span>, we analyze how this linear process convergence is transferred to the solution of the semilinear equation. We illustrate the ideas in two examples. First a reaction-diffusion equation in a bounded smooth domain <span>(Omega subset {mathbb {R}}^{3})</span></p><span>$$begin{aligned}begin{aligned}&amp;(u_{varepsilon })_t - div (a_{varepsilon } (t,x) nabla u_{varepsilon }) +u_{varepsilon } = f_{varepsilon } (t,u_{varepsilon }), quad xin Omega , t&gt; tau , end{aligned} end{aligned}$$</span><p>where <span>(a_varepsilon )</span> converges to a function <span>(a_0)</span>, <span>(f_{varepsilon })</span> converges to <span>(f_0)</span>. We apply the abstract theory in this example, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated with each problem. The second example is a nonautonomous strongly damped wave equation </p><span>$$begin{aligned} u_{tt}+(-a(t) Delta _D) u + 2 (-a(t)Delta _D)^{frac{1}{2}} u_t = f(t,u), quad xin Omega , t&gt;tau ,end{aligned}$$</span><p>where <span>(Delta _D)</span> is the Laplacian operator with Dirichlet boundary conditions in a domain <span>(Omega )</span> and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"204 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592522","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}

{"title":"Sharp ill-posedness for the Hunter–Saxton equation on the line","authors":"","doi":"10.1007/s00028-024-00962-x","DOIUrl":"https://doi.org/10.1007/s00028-024-00962-x","url":null,"abstract":"<h3>Abstract</h3> <p>The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space <span> <span>(mathcal {B})</span> </span> is constructed to overcome this difficulty. More precisely, if the initial data <span> <span>(u_0in mathcal {B}cap dot{H}^{1}(mathbb {R}),)</span> </span> the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if <span> <span>(u_0in mathcal {B})</span> </span> but <span> <span>(u_0notin dot{H}^{1}(mathbb {R}),)</span> </span> the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in <span> <span>(L^{infty }(0,T;L^{infty }(mathbb {R})))</span> </span> must have the <span> <span>(dot{H}^1)</span> </span> norm.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"238 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571839","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}

{"title":"Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces","authors":"Effie Papageorgiou","doi":"10.1007/s00028-024-00959-6","DOIUrl":"https://doi.org/10.1007/s00028-024-00959-6","url":null,"abstract":"<p>This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces <i>G</i>/<i>K</i> of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for <span>(L^1)</span> initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-<i>K</i>-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-<i>K</i>-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on <i>G</i>/<i>K</i>. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as <span>(L^1)</span> asymptotic convergence without the assumption of bi-<i>K</i>-invariance.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"7 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592737","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}

{"title":"KFP operators with coefficients measurable in time and Dini continuous in space","authors":"","doi":"10.1007/s00028-024-00964-9","DOIUrl":"https://doi.org/10.1007/s00028-024-00964-9","url":null,"abstract":"<h3>Abstract</h3> <p>We consider degenerate Kolmogorov–Fokker–Planck operators <span> <span>$$begin{aligned} mathcal {L}u&amp;=sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}} ^{2}u+sum _{k,j=1}^{N}b_{jk}x_{k}partial _{x_{j}}u-partial _{t}u&amp;equiv sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}}^{2}u+Yu end{aligned}$$</span> </span>(with <span> <span>((x,t)in mathbb {R}^{N+1})</span> </span> and <span> <span>(1le m_{0}le N)</span> </span>) such that the corresponding model operator having constant <span> <span>(a_{ij})</span> </span> is hypoelliptic, translation invariant w.r.t. a Lie group operation in <span> <span>(mathbb {R} ^{N+1})</span> </span> and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix <span> <span>((a_{ij})_{i,j=1}^{m_{0}})</span> </span> is symmetric and uniformly positive on <span> <span>(mathbb {R}^{m_{0}})</span> </span>. The coefficients <span> <span>(a_{ij})</span> </span> are bounded and <em>Dini continuous in space</em>, and only bounded measurable in time. This means that, setting <span> <span>$$begin{aligned} mathrm {(i)}&amp;,,S_{T}=mathbb {R}^{N}times left( -infty ,Tright) , mathrm {(ii)}&amp;,,omega _{f,S_{T}}(r) = sup _{begin{array}{c} (x,t),(y,t)in S_{T} Vert x-yVert le r end{array}}vert f(x,t) -f(y,t)vert mathrm {(iii)}&amp;,,Vert fVert _{mathcal {D}( S_{T}) } =int _{0}^{1} frac{omega _{f,S_{T}}(r) }{r}dr+Vert fVert _{L^{infty }left( S_{T}right) } end{aligned}$$</span> </span>we require the finiteness of <span> <span>(Vert a_{ij}Vert _{mathcal {D}(S_{T})})</span> </span>. We bound <span> <span>(omega _{u_{x_{i}x_{j}},S_{T}})</span> </span>, <span> <span>(Vert u_{x_{i}x_{j}}Vert _{L^{infty }( S_{T}) })</span> </span> (<span> <span>(i,j=1,2,...,m_{0})</span> </span>), <span> <span>(omega _{Yu,S_{T}})</span> </span>, <span> <span>(Vert YuVert _{L^{infty }( S_{T}) })</span> </span> in terms of <span> <span>(omega _{mathcal {L}u,S_{T}})</span> </span>, <span> <span>(Vert mathcal {L}uVert _{L^{infty }( S_{T}) })</span> </span> and <span> <span>(Vert uVert _{L^{infty }left( S_{T}right) })</span> </span>, getting a control on the uniform continuity in space of <span> <span>(u_{x_{i}x_{j}},Yu)</span> </span> if <span> <span>(mathcal {L}u)</span> </span> is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients <span> <span>(a_{ij})</span> </span> and <span> <span>(mathcal {L}u)</span> </span> are log-Dini continuous, meaning the finiteness of the quantity <span> <span>$$begin{aligned} int _{0}^{1}frac{omega _{f,S_{T}}left( rright) }{r}left| log rright| dr, end{aligned}$$</span> </span>we prove that <span> <span>(u_{x_{i}x_{j}})</span> </span> and <em>Yu</em> are Dini continuous; moreover, in this case, the derivatives <span> <span>(u_{x_{i}x_{j}})</span> </span> are locally uniformly continuous in space <em>and time</em>. </p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571932","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}

{"title":"A perturbative approach to Hölder continuity of solutions to a nonlocal p-parabolic equation","authors":"Alireza Tavakoli","doi":"10.1007/s00028-024-00949-8","DOIUrl":"https://doi.org/10.1007/s00028-024-00949-8","url":null,"abstract":"<p>We study local boundedness and Hölder continuity of a parabolic equation involving the fractional <i>p</i>-Laplacian of order <i>s</i>, with <span>(0&lt;s&lt;1)</span>, <span>(2le p &lt; infty )</span>, with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"40 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151272","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}

{"title":"On the high friction limit for the complete Euler system","authors":"Eduard Feireisl, Piotr Gwiazda, Young-Sam Kwon, Agnieszka Świerczewska-Gwiazda","doi":"10.1007/s00028-024-00956-9","DOIUrl":"https://doi.org/10.1007/s00028-024-00956-9","url":null,"abstract":"<p>We show that solutions of the complete Euler system of gas dynamics perturbed by a friction term converge to a solution of the porous medium equation in the high friction/long time limit. The result holds in the largest possible class of generalized solutions–the measure–valued solutions of the Euler system.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"23 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153537","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}

{"title":"Generic alignment conjecture for systems of Cucker–Smale type","authors":"Roman Shvydkoy","doi":"10.1007/s00028-024-00950-1","DOIUrl":"https://doi.org/10.1007/s00028-024-00950-1","url":null,"abstract":"<p>The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker–Smale system with a strictly local communication align to the common mean velocity. In this note, we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case, the conjecture is proved when <i>N</i>, the number of agents, is equal to 2. It follows from a more general result, stating that for a system of any size for almost every data at least two agents align. The analysis is extended to the open space <span>(mathbb {R}^n)</span> in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"69 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153547","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}

{"title":"The Stokes Dirichlet-to-Neumann operator","authors":"C. Denis, A. F. M. ter Elst","doi":"10.1007/s00028-023-00930-x","DOIUrl":"https://doi.org/10.1007/s00028-023-00930-x","url":null,"abstract":"<p>Let <span>(Omega subset mathbb {R}^d)</span> be a bounded open connected set with Lipschitz boundary. Let <span>(A^N)</span> and <span>(A^D)</span> be the Stokes Neumann operator and Stokes Dirichlet operator on <span>(Omega )</span>, respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"51 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153548","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}

{"title":"Refined decay rates of $$C_0$$ -semigroups on Banach spaces","authors":"Genilson Santana, Silas L. Carvalho","doi":"10.1007/s00028-024-00957-8","DOIUrl":"https://doi.org/10.1007/s00028-024-00957-8","url":null,"abstract":"<p>We study rates of decay for <span>(C_0)</span>-semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with <span>(|s|^{beta }log (|s|)^b)</span>, <span>(beta , b ge 0)</span>, as <span>(|s|rightarrow infty )</span>, and with <span>(|s|^{-alpha }log (1/|s|)^a)</span>, <span>(alpha , a ge 0)</span>, as <span>(|s|rightarrow 0)</span>. Our results do not suppose that the semigroup is bounded. In particular, for <span>(a=b=0)</span>, our results improve the rates involving Fourier types obtained by Rozendaal and Veraar (J Funct Anal 275(10):2845–2894, 2018).</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"11 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153455","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}

{"title":"Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation","authors":"Wilberclay G. Melo","doi":"10.1007/s00028-024-00947-w","DOIUrl":"https://doi.org/10.1007/s00028-024-00947-w","url":null,"abstract":"<p>Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG) </p><span>$$begin{aligned} theta _t ;!+, (-Delta )^{alpha },theta ,+, u_{theta } cdot nabla theta ;=; 0, quad hbox {with } u_{theta };=;(partial _2(-Delta )^{frac{gamma -2}{2}}theta , -partial _1(-Delta )^{frac{gamma -2}{2}}theta ), end{aligned}$$</span><p>in the non-homogenous Sobolev space <span>(H^{1+gamma -2alpha }(mathbb {R}^2))</span>, where <span>(alpha in (0,frac{1}{2}))</span> and <span>(gamma in (1,2alpha +1))</span>. To this end, we need consider that the initial data for this equation are small. More precisely, we assume that <span>(Vert theta _0Vert _{H^{1+gamma -2alpha }})</span> is small enough in order to obtain a unique <span>(theta in C([0,infty );H^{1+gamma -2alpha }(mathbb {R}^2)))</span> that solves (MQG) and satisfies the following limit: </p><span>$$begin{aligned} lim _{trightarrow infty } Vert theta (t)Vert _{H^{1+gamma -2alpha }}=0. end{aligned}$$</span>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"23 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153544","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}