Journal of Evolution Equations最新文献

{"title":"Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case","authors":"Sebastian Grube","doi":"10.1007/s00028-024-00970-x","DOIUrl":"https://doi.org/10.1007/s00028-024-00970-x","url":null,"abstract":"<p>We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending <i>explicitly</i> on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution <span>(uin L^1cap L^infty )</span>. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities <i>u</i>. In particular, every weak solution of this equation with time marginal law densities <i>u</i> can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities <i>u</i>.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"50 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805879","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":"Propagation of anisotropic Gabor singularities for Schrödinger type equations","authors":"","doi":"10.1007/s00028-024-00963-w","DOIUrl":"https://doi.org/10.1007/s00028-024-00963-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity <span> <span>(a(lambda x, lambda ^sigma xi ) = lambda ^{1+sigma } a(x,xi ))</span> </span> for <span> <span>(lambda &gt; 0)</span> </span> where <span> <span>(sigma &gt; 0)</span> </span> is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"298 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592728","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":"Large deviation principle for multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motions","authors":"Guangjun Shen, Huan Zhou, Jiang-Lun Wu","doi":"10.1007/s00028-024-00960-z","DOIUrl":"https://doi.org/10.1007/s00028-024-00960-z","url":null,"abstract":"<p>In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index <span>(H&gt;frac{1}{2})</span>) and standard Brownian motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592195","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":"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":"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":"Some aspects of the Floquet theory for the heat equation in a periodic domain","authors":"Marcus Rosenberg, Jari Taskinen","doi":"10.1007/s00028-024-00951-0","DOIUrl":"https://doi.org/10.1007/s00028-024-00951-0","url":null,"abstract":"<p>We treat the linear heat equation in a periodic waveguide <span>(Pi subset {{mathbb {R}}}^d)</span>, with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform <span>({{textsf{F}}})</span> to the equation yields a heat equation with mixed boundary conditions on the periodic cell <span>(varpi )</span> of <span>(Pi )</span>, and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces <span>({{mathcal {H}}}_S subset L^2(Pi ))</span> corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in <span>({{mathcal {H}}}_S)</span>.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"2012 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150839","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 separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions","authors":"Andrea Giorgini","doi":"10.1007/s00028-024-00953-y","DOIUrl":"https://doi.org/10.1007/s00028-024-00953-y","url":null,"abstract":"<p>We consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. This model describes phase separation in binary fluid mixtures. Given any global solution (whose existence and uniqueness are already known), we prove the so-called <i>instantaneous</i> and <i>uniform</i> separation property: any global solution with initial finite energy is globally confined (in the <span>(L^infty )</span> metric) in the interval <span>([-1+delta ,1-delta ])</span> on the time interval <span>([tau ,infty ))</span> for any <span>(tau &gt;0)</span>, where <span>(delta )</span> only depends on the norms of the initial datum, <span>(tau )</span> and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"43 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151059","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":"Bi-objective and hierarchical control for the Burgers equation","authors":"F. D. Araruna, E. Fernández-Cara, L. C. da Silva","doi":"10.1007/s00028-024-00952-z","DOIUrl":"https://doi.org/10.1007/s00028-024-00952-z","url":null,"abstract":"<p>We present some results concerning the control of the Burgers equation. We analyze a bi-objective optimal control problem and then the hierarchical null controllability through a Stackelberg–Nash strategy, with one leader and two followers. The results may be viewed as an extension to this nonlinear setting of a previous analysis performed for linear and semilinear heat equations. They can also be regarded as a first step in the solution of control problems of this kind for the Navier–Stokes equations.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"34 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153458","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}