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Analysis of PDEs
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class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AP/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 16 of 16 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2503.20922" title="Abstract" id="2503.20922"> arXiv:2503.20922 </a> [<a href="/pdf/2503.20922" title="Download PDF" id="pdf-2503.20922" aria-labelledby="pdf-2503.20922">pdf</a>, <a href="https://arxiv.org/html/2503.20922v1" title="View HTML" id="html-2503.20922" aria-labelledby="html-2503.20922" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.20922" title="Other formats" id="oth-2503.20922" aria-labelledby="oth-2503.20922">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A kinetic theory approach to consensus formation in financial markets </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Attali,+J">Jean-Gabriel Attali</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Salvarani,+F">Francesco Salvarani</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> We investigate the relationship between analysts' one-year consensus forecasts for the S&P 500 index and its current level. Contrary to the conventional view that sentiment drives price forecasts, our analysis predicts that the average consensus has no effect on forecasts, while the current S&P 500 index level alone is sufficient to anticipate analysts' price expectations. <br>Employing a kinetic theory framework, we model the dynamics of analysts' opinions, by taking into account both the mutual influences shaping price consensus and the dynamics of the actual S&P 500 index level. Testing the model on real data on a long-time series of data shows that just three free parameters are enough to accurately describe the one-year average price forecasts of analysts. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2503.20955" title="Abstract" id="2503.20955"> arXiv:2503.20955 </a> [<a href="/pdf/2503.20955" title="Download PDF" id="pdf-2503.20955" aria-labelledby="pdf-2503.20955">pdf</a>, <a href="https://arxiv.org/html/2503.20955v1" title="View HTML" id="html-2503.20955" aria-labelledby="html-2503.20955" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.20955" title="Other formats" id="oth-2503.20955" aria-labelledby="oth-2503.20955">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Propagation of Shubin-Sobolev singularities of Weyl-quantizations of complex quadratic forms </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Malagutti,+M">Marcello Malagutti</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Parmeggiani,+A">Alberto Parmeggiani</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Tramontana,+D">Davide Tramontana</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> The aim of this work is to develop the H枚rmander microlocal theory in the isotropic framework and use the results we obtain to study the propagation of singularities for an evolution problem, with diffusive part given by a Weyl-quantization of a complex quadratic form on the phase space. </p> </div> </dd> <dt> <a name='item3'>[3]</a> <a href ="/abs/2503.20971" title="Abstract" id="2503.20971"> arXiv:2503.20971 </a> [<a href="/pdf/2503.20971" title="Download PDF" id="pdf-2503.20971" aria-labelledby="pdf-2503.20971">pdf</a>, <a href="/format/2503.20971" title="Other formats" id="oth-2503.20971" aria-labelledby="oth-2503.20971">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Local well-posedness for cubic fractional Schr枚dinger equations with derivatives on the right-hand side </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Dughayshim,+A">Ahmed Dughayshim</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Farina,+S+R">Silvino Reyes Farina</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Schikorra,+A">Armin Schikorra</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$. <br>This equation is a model equation for for $s$-Schr枚dinger map equation \[ \partial_t \psi = \psi \wedge (-\Delta)^s \psi: \quad \psi: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \] </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2503.21131" title="Abstract" id="2503.21131"> arXiv:2503.21131 </a> [<a href="/pdf/2503.21131" title="Download PDF" id="pdf-2503.21131" aria-labelledby="pdf-2503.21131">pdf</a>, <a href="https://arxiv.org/html/2503.21131v1" title="View HTML" id="html-2503.21131" aria-labelledby="html-2503.21131" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21131" title="Other formats" id="oth-2503.21131" aria-labelledby="oth-2503.21131">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Long-time dynamics of a parabolic-ODE SIS epidemic model with saturated incidence mechanism </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Peng,+R">Rui Peng</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Salako,+R">Rachidi Salako</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+Y">Yixiang Wu</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Dynamical Systems (math.DS) </div> <p class='mathjax'> In this paper, we investigate a parabolic-ODE SIS epidemic model with no-flux boundary conditions in a heterogeneous environment. The model incorporates a saturated infection mechanism \({SI}/(m(x) + S + I)\) with \(m \geq,\,\not\equiv 0\). This study is motivated by disease control strategies, such as quarantine and lockdown, that limit population movement. We examine two scenarios: one where the movement of the susceptible population is restricted, and another where the movement of the infected population is neglected. We establish the long-term dynamics of the solutions in each scenario. Compared to previous studies that assume the absence of a saturated incidence function (i.e., $m\equiv 0$), our findings highlight the novel and significant interplay between total population size, transmission risk level, and the saturated incidence function in influencing disease persistence, extinction, and spatial distribution. Numerical simulations are performed to validate the theoretical results, and the implications of the results are discussed in the context of disease control and eradication strategies. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2503.21181" title="Abstract" id="2503.21181"> arXiv:2503.21181 </a> [<a href="/pdf/2503.21181" title="Download PDF" id="pdf-2503.21181" aria-labelledby="pdf-2503.21181">pdf</a>, <a href="https://arxiv.org/html/2503.21181v1" title="View HTML" id="html-2503.21181" aria-labelledby="html-2503.21181" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21181" title="Other formats" id="oth-2503.21181" aria-labelledby="oth-2503.21181">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Subwavelength Phononic Bandgaps in High-Contrast Elastic Media </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ren,+Y">Yuanchun Ren</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+B">Bochao Chen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Gao,+Y">Yixian Gao</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Li,+P">Peijun Li</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 25pp </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> Inspired by [25], this paper investigates subwavelength bandgaps in phononic crystals consisting of periodically arranged hard elastic materials embedded in a soft elastic background medium. Our contributions are threefold. First, we introduce the quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form to characterize the subwavelength resonant frequencies, which are identified through the condition that the determinant of a certain matrix vanishes. Second, we derive asymptotic expansions for these resonant frequencies and the corresponding non-trivial solutions, thereby establishing the existence of subwavelength phononic bandgaps in elastic media. Finally, we analyze dilute structures in three dimensions, where the spacing between adjacent resonators is significantly larger than the characteristic size of an individual resonator, allowing the inter-resonator interactions to be neglected. In particular, an illustrative example is presented in which the resonator is modeled as a ball. </p> </div> </dd> <dt> <a name='item6'>[6]</a> <a href ="/abs/2503.21245" title="Abstract" id="2503.21245"> arXiv:2503.21245 </a> [<a href="/pdf/2503.21245" title="Download PDF" id="pdf-2503.21245" aria-labelledby="pdf-2503.21245">pdf</a>, <a href="/format/2503.21245" title="Other formats" id="oth-2503.21245" aria-labelledby="oth-2503.21245">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chan,+H">Hardy Chan</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Fern%C3%A1ndez-Real,+X">Xavier Fern谩ndez-Real</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Figalli,+A">Alessio Figalli</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Serra,+J">Joaquim Serra</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in \(\mathbb{R}^{n+1}\) is one-dimensional if \(n \leq 7\). A stronger version of the conjecture, also widely studied and often called ``the stable De Giorgi conjecture'', proposes that every stable solution in \(\mathbb{R}^n\) must be one-dimensional for \(n \leq 7\). To this date, both conjectures remain open for \(3 \leq n \leq 7\). <br>An elegant variant of this problem, advocated by Caffarelli, C贸rdoba, and Jerison since the 1990s, considers a free boundary version of the Allen--Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen--Cahn equation. <br>In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 for the free boundary Allen--Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4. To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory -- after significant refinements -- to the free boundary Allen--Cahn equation. </p> </div> </dd> <dt> <a name='item7'>[7]</a> <a href ="/abs/2503.21247" title="Abstract" id="2503.21247"> arXiv:2503.21247 </a> [<a href="/pdf/2503.21247" title="Download PDF" id="pdf-2503.21247" aria-labelledby="pdf-2503.21247">pdf</a>, <a href="https://arxiv.org/html/2503.21247v1" title="View HTML" id="html-2503.21247" aria-labelledby="html-2503.21247" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21247" title="Other formats" id="oth-2503.21247" aria-labelledby="oth-2503.21247">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Remarks on the commutation relations between the Gauss--Weierstrass semigroup and monomial weights </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Huang,+Y+C">Yi C. Huang</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Kusaba,+R">Ryunosuke Kusaba</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ozawa,+T">Tohru Ozawa</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 16 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA) </div> <p class='mathjax'> We consider the commutation relations between the Gauss--Weierstrass semigroup on $\mathbb{R}^{n}$, generated by convolution with the complex Gauss kernel, and monomial weights. We provide an explicit representation and a sharper estimate of the commutation relations with more concise proofs than those in previous works. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2503.21299" title="Abstract" id="2503.21299"> arXiv:2503.21299 </a> [<a href="/pdf/2503.21299" title="Download PDF" id="pdf-2503.21299" aria-labelledby="pdf-2503.21299">pdf</a>, <a href="https://arxiv.org/html/2503.21299v1" title="View HTML" id="html-2503.21299" aria-labelledby="html-2503.21299" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21299" title="Other formats" id="oth-2503.21299" aria-labelledby="oth-2503.21299">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Microscopic limits of PDEs modeling macroscopic heat conduction </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Mickens,+R">Ronald Mickens</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Washington,+T">Talitha Washington</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 10 pages, 1 figure </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> We show that it is possible to construct microscopic-level discrete equations from macroscopic modeling PDEs for heat conduction in one space dimension. The significance of this result is that, in general, one starts from microscopic theories and then take their continuum limits to obtain the corresponding macroscopic PDEs, whereas here it is demonstrated that the reverse procedure is also possible. While our focus is on heat conduction, we discuss the applicability of our methodology to other physical systems. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2503.21509" title="Abstract" id="2503.21509"> arXiv:2503.21509 </a> [<a href="/pdf/2503.21509" title="Download PDF" id="pdf-2503.21509" aria-labelledby="pdf-2503.21509">pdf</a>, <a href="/format/2503.21509" title="Other formats" id="oth-2503.21509" aria-labelledby="oth-2503.21509">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Nonlinear Stability of Large-Period Traveling Waves Bifurcating from the Heteroclinic Loop in the FitzHugh-Nagumo Equation </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Li,+J">Ji Li</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wang,+K">Ke Wang</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+Q">Qiliang Wu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Yu,+Q">Qing Yu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 45 pages,4 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Dynamical Systems (math.DS) </div> <p class='mathjax'> A wave front and a wave back that spontaneously connect two hyperbolic equilibria, known as a heteroclinic wave loop, give rise to periodic waves with arbitrarily large spatial periods through the heteroclinic bifurcation. The nonlinear stability of these periodic waves is established in the setting of the FitzHugh-Nagumo equation, which is a well-known reaction-diffusion model with degenerate diffusion. First, for general systems, we give the expressions of spectra with small modulus for linearized operators about these periodic waves via the Lyapunov-Schmidt reduction and the Lin-Sandstede method. Second, applying these spectral results to the FitzHugh-Nagumo equation, we establish their diffusive spectral stability. Finally, we consider the nonlinear stability of these periodic waves against localized perturbations. We introduce a spatiotemporal phase modulation $\varphi$, and couple it with the associated modulated perturbation $\mathbf{V}$ along with the unmodulated perturbation $\mathbf{\widetilde{V}}$ to close a nonlinear iteration argument. </p> </div> </dd> <dt> <a name='item10'>[10]</a> <a href ="/abs/2503.21523" title="Abstract" id="2503.21523"> arXiv:2503.21523 </a> [<a href="/pdf/2503.21523" title="Download PDF" id="pdf-2503.21523" aria-labelledby="pdf-2503.21523">pdf</a>, <a href="/format/2503.21523" title="Other formats" id="oth-2503.21523" aria-labelledby="oth-2503.21523">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A weak energy identity for $(n+伪)$-harmonic maps with a free boundary in a sphere </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Martino,+D">Dorian Martino</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mazowiecka,+K">Katarzyna Mazowiecka</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Rodiac,+R">R茅my Rodiac</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> In this article, we show that sequences of $(n+\alpha)$-harmonic maps with a free boundary in $\mathbb S^{d-1}$, where $\alpha$ is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the limiting energy is equal to the energy of the macroscopic limit plus the sum of the energies of certain ``bubbles'', each multiplied by a corresponding coefficient. </p> </div> </dd> <dt> <a name='item11'>[11]</a> <a href ="/abs/2503.21527" title="Abstract" id="2503.21527"> arXiv:2503.21527 </a> [<a href="/pdf/2503.21527" title="Download PDF" id="pdf-2503.21527" aria-labelledby="pdf-2503.21527">pdf</a>, <a href="https://arxiv.org/html/2503.21527v1" title="View HTML" id="html-2503.21527" aria-labelledby="html-2503.21527" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21527" title="Other formats" id="oth-2503.21527" aria-labelledby="oth-2503.21527">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Dispersive estimates and optimality for Schr枚dinger equations on product cones </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Taira,+K">Kouichi Taira</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 19 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> In this paper, we study time decay estimates for the Schr枚dinger propagator on the product cone $(X,g)$, where $X=C(\rho \mathbb{S}^{n-1})=(0,\infty)\times \rho\mathbb{S}^{n-1}$. We prove that the usual dispersive estimate holds when the radius $\rho$ is greater than or equal to 1 and fails otherwise. A part of the former result was already established in a recent paper by Jia-Zhang. The method used here relies purely on harmonic analysis, whereas Jia-Zhang employed microlocal analysis to capture the precise asymptotic behavior of the propagator. </p> </div> </dd> <dt> <a name='item12'>[12]</a> <a href ="/abs/2503.21580" title="Abstract" id="2503.21580"> arXiv:2503.21580 </a> [<a href="/pdf/2503.21580" title="Download PDF" id="pdf-2503.21580" aria-labelledby="pdf-2503.21580">pdf</a>, <a href="https://arxiv.org/html/2503.21580v1" title="View HTML" id="html-2503.21580" aria-labelledby="html-2503.21580" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21580" title="Other formats" id="oth-2503.21580" aria-labelledby="oth-2503.21580">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=B%C3%A4uerlein,+F">Fabian B盲uerlein</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ricc%C3%B2,+S">Samuele Ricc貌</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sch%C3%A4tzler,+L">Leah Sch盲tzler</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> We prove global higher integrability for functionals of double-phase type under a uniform local capacity density condition on the complement of the considered domain $\Omega \subset \mathbb{R}^n$. In this context, we investigate a new natural notion of variational capacity associated to the double-phase integrand. Under the related fatness condition for the complement of $\Omega$, we establish an integral Hardy inequality. Further, we show that fatness of $\mathbb{R}^n \setminus \Omega$ is equivalent to a boundary Poincar茅 inequality, a pointwise Hardy inequality and to the local uniform $p$-fatness of $\mathbb{R}^n \setminus \Omega$. We provide a counterexample that shows that the expected Maz'ya type inequality - a key intermediate step toward global higher integrability - does not hold with the notion of capacity involving the double-phase functional itself. </p> </div> </dd> <dt> <a name='item13'>[13]</a> <a href ="/abs/2503.21604" title="Abstract" id="2503.21604"> arXiv:2503.21604 </a> [<a href="/pdf/2503.21604" title="Download PDF" id="pdf-2503.21604" aria-labelledby="pdf-2503.21604">pdf</a>, <a href="https://arxiv.org/html/2503.21604v1" title="View HTML" id="html-2503.21604" aria-labelledby="html-2503.21604" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21604" title="Other formats" id="oth-2503.21604" aria-labelledby="oth-2503.21604">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the dynamics of leapfrogging vortex rings </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Donati,+M">Martin Donati</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Hientzsch,+L+E">Lars Eric Hientzsch</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Lacave,+C">Christophe Lacave</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Miot,+E">Evelyne Miot</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Mathematical Physics (math-ph) </div> <p class='mathjax'> The evolution of highly concentrated vorticity around rings in the three-dimensional axisymmetric Euler equations is studied in a regime for which the leapfrogging dynamics predicted by Helmholtz is expected to occur. For a general class of initial data corresponding to vortex rings of sufficiently small thickness $\varepsilon>0$, we prove weak and strong localization estimates for the vortex rings at positive times up to some time $T_0/|\ln \varepsilon|$ for fixed $T_0$, independent of $\varepsilon$. Moreover, we derive the motion law for the position of the rings on $[0,T_0/|\ln \varepsilon|]$. To the best of our knowledge, this is the first result of a rigorous derivation of the leapfrogging motion up to times of order $|\ln\varepsilon|^{-1}$ for general initial data and the suitable scaling regime. The singular interaction of rings requires significant improvements of weak and strong localization estimates obtained in prior works. Our method is based on the combination of a new variational argument and a recently introduced double iterative procedure. Possibly due to natural filamentation of the vortex rings, which induces a lack of control of the support of the solution in the symmetry axis direction, our method does not allow to reach the time at which the rings have performed a full rotation around each other. </p> </div> </dd> <dt> <a name='item14'>[14]</a> <a href ="/abs/2503.21625" title="Abstract" id="2503.21625"> arXiv:2503.21625 </a> [<a href="/pdf/2503.21625" title="Download PDF" id="pdf-2503.21625" aria-labelledby="pdf-2503.21625">pdf</a>, <a href="https://arxiv.org/html/2503.21625v1" title="View HTML" id="html-2503.21625" aria-labelledby="html-2503.21625" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21625" title="Other formats" id="oth-2503.21625" aria-labelledby="oth-2503.21625">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the reverse isoperimetric inequality in Gauss space </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Brock,+F">Friedemann Brock</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Chiacchio,+F">Francesco Chiacchio</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects of the reverse problem have not yet been investigated. In particular, to the best of our knowledge, there seem to be no results on the shape that the isoperimetric set should take. Here, through a local perturbation analysis, we show that smooth perimeter-maximizing sets have locally flat boundaries. Additionally, we derive sharper perimeter bounds than those previously known, particularly for specific classes of convex sets such as the convex sets symmetric with respect to the axes. Finally, for quadrilaterals with vertices on the coordinate axes, we prove that the set maximizing the perimeter "degenerates" into the x-axis, traversed twice. </p> </div> </dd> <dt> <a name='item15'>[15]</a> <a href ="/abs/2503.21684" title="Abstract" id="2503.21684"> arXiv:2503.21684 </a> [<a href="/pdf/2503.21684" title="Download PDF" id="pdf-2503.21684" aria-labelledby="pdf-2503.21684">pdf</a>, <a href="https://arxiv.org/html/2503.21684v1" title="View HTML" id="html-2503.21684" aria-labelledby="html-2503.21684" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21684" title="Other formats" id="oth-2503.21684" aria-labelledby="oth-2503.21684">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Decorated phases in triblock copolymers: zeroth- and first-order analysis </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Alama,+S">Stanley Alama</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Bronsard,+L">Lia Bronsard</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Lu,+X">Xinyang Lu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wang,+C">Chong Wang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> We study a two-dimensional inhibitory ternary system characterized by a free energy functional which combines an interface short-range interaction energy promoting micro-domain growth with a Coulomb-type long-range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a scenario in which two species are dominant and one species is vanishingly small. In this scenario two energy levels are distinguished: the zeroth-order energy encodes information on the optimal arrangement of the dominant constituents, while the first-order energy gives the shape of the vanishing constituent. This first-order energy also shows that, for any optimal configuration, the vanishing phase must lie on the boundary between the two dominant constituents and form lens clusters also known as vesica piscis. </p> </div> </dd> <dt> <a name='item16'>[16]</a> <a href ="/abs/2503.21700" title="Abstract" id="2503.21700"> arXiv:2503.21700 </a> [<a href="/pdf/2503.21700" title="Download PDF" id="pdf-2503.21700" aria-labelledby="pdf-2503.21700">pdf</a>, <a href="https://arxiv.org/html/2503.21700v1" title="View HTML" id="html-2503.21700" aria-labelledby="html-2503.21700" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21700" title="Other formats" id="oth-2503.21700" aria-labelledby="oth-2503.21700">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Barbera,+D">Daniele Barbera</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Boni,+F">Filippo Boni</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Dovetta,+S">Simone Dovetta</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Tentarelli,+L">Lorenzo Tentarelli</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 25 pages, 1 figure. Keywords: doubly nonlinear Schr枚dinger, nonlinear point interactions, normalized solutions, ground states </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Mathematical Physics (math-ph) </div> <p class='mathjax'> We investigate normalized solutions for doubly nonlinear Schr枚dinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $\delta$--type at the origin. We provide a complete characterization of existence and uniqueness for normalized solutions and for energy ground states for every value of the nonlinearity powers. We show that the interplay between a defocusing standard and a focusing point nonlinearity gives rise to new phenomena with respect to those observed with single nonlinearities, standard combined nonlinearities, and combined focusing standard and pointwise nonlinearities. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 4 of 4 entries)</h3> <dt> <a name='item17'>[17]</a> <a href ="/abs/2503.20809" title="Abstract" id="2503.20809"> arXiv:2503.20809 </a> (cross-list from math.FA) [<a href="/pdf/2503.20809" title="Download PDF" id="pdf-2503.20809" aria-labelledby="pdf-2503.20809">pdf</a>, <a href="https://arxiv.org/html/2503.20809v1" title="View HTML" id="html-2503.20809" aria-labelledby="html-2503.20809" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.20809" title="Other formats" id="oth-2503.20809" aria-labelledby="oth-2503.20809">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Limiting Behaviors of Besov Seminorms for Dunkl Operators </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Li,+H">Huaiqian Li</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+B">Bingyao Wu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 29 pp, 1 figuer; submitted </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Functional Analysis (math.FA)</span>; Analysis of PDEs (math.AP); Probability (math.PR) </div> <p class='mathjax'> As $s\rightarrow0^+$, we establish limiting formulas of Besov seminorms and nonlocal perimeters associated with the Dunkl operator, a (nonlocal) differential-difference operator parameterized by multiplicity functions and finite reflection groups. Our results are further developments of both the Maz'ya--Shaposhnikova limiting formula for the Gagliardo seminorm and the asymptotic behavior of the (relative) fractional $s$-perimeter. The main contribution is twofold. On the one hand, to establish our dimension-free Maz'ya--Shaposhnikova limiting formula, we develop a simplified approach which do not depend on the density property of the corresponding Besov space and turns out to be quite robust. On the other hand, to derive the limiting formula of our nonlocal perimeter, we do not demand additional regularity on the (topological) boundary of the domain, and to obtain the converse assertion, our assumption on the boundary regularity of the domain, which allows for fractals, is much weaker than those in existing literatures. </p> </div> </dd> <dt> <a name='item18'>[18]</a> <a href ="/abs/2503.21405" title="Abstract" id="2503.21405"> arXiv:2503.21405 </a> (cross-list from math-ph) [<a href="/pdf/2503.21405" title="Download PDF" id="pdf-2503.21405" aria-labelledby="pdf-2503.21405">pdf</a>, <a href="https://arxiv.org/html/2503.21405v1" title="View HTML" id="html-2503.21405" aria-labelledby="html-2503.21405" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21405" title="Other formats" id="oth-2503.21405" aria-labelledby="oth-2503.21405">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the relativistic effect in the Dirac--Fock theory </div> <div class='list-authors'><a href="https://arxiv.org/search/math-ph?searchtype=author&query=Meng,+L">Long Meng</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Mathematical Physics (math-ph)</span>; Analysis of PDEs (math.AP); Atomic and Molecular Clusters (physics.atm-clus) </div> <p class='mathjax'> In this paper, we study the error bound of the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy. This error bound is called the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\cO(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the leading order relativistic correction, which comprises the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. The proof is based on a delicate study of projections onto the positive eigenspace of some Dirac operators. <br>To our knowledge, it is the first mathematical derivation of the leading order relativistic correction for nonlinear Dirac ground-state energies. Our method paves the way to study the relativistic effects in general nonlinear Dirac problems. </p> </div> </dd> <dt> <a name='item19'>[19]</a> <a href ="/abs/2503.21572" title="Abstract" id="2503.21572"> arXiv:2503.21572 </a> (cross-list from math.PR) [<a href="/pdf/2503.21572" title="Download PDF" id="pdf-2503.21572" aria-labelledby="pdf-2503.21572">pdf</a>, <a href="https://arxiv.org/html/2503.21572v1" title="View HTML" id="html-2503.21572" aria-labelledby="html-2503.21572" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21572" title="Other formats" id="oth-2503.21572" aria-labelledby="oth-2503.21572">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Convergence of a Stochastic Particle System to the Continuous Generalized Exchange-Driven Growth Model </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Lam,+C+Y">Chun Yin Lam</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Schlichting,+A">Andr茅 Schlichting</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Mathematical Physics (math-ph); Analysis of PDEs (math.AP) </div> <p class='mathjax'> The continuous generalized exchange-driven growth model (CGEDG) is a system of integro-differential equations describing the evolution of cluster mass under mass exchange. The rate of exchange depends on the masses of the clusters involved and the mass being exchanged. This can be viewed as both a continuous generalization of the exchange-driven growth model and a coagulation-fragmentation equation that generalizes the continuous Smoluchowski equation. <br>Starting from a Markov jump process that describes a finite stochastic interacting particle system with exchange dynamics, we prove the weak law of large numbers for this process for sublinearly growing kernels in the mean-field limit. We establish the tightness of the stochastic process on a measure-valued Skorokhod space induced by the $1$-Wasserstein metric, from which we deduce the existence of solutions to the (CGEDG) system. The solution is shown to have a Lebesgue density under suitable assumptions on the initial data. Moreover, within the class of solutions with density, we establish the uniqueness under slightly more restrictive conditions on the kernel. </p> </div> </dd> <dt> <a name='item20'>[20]</a> <a href ="/abs/2503.21675" title="Abstract" id="2503.21675"> arXiv:2503.21675 </a> (cross-list from nlin.SI) [<a href="/pdf/2503.21675" title="Download PDF" id="pdf-2503.21675" aria-labelledby="pdf-2503.21675">pdf</a>, <a href="https://arxiv.org/html/2503.21675v1" title="View HTML" id="html-2503.21675" aria-labelledby="html-2503.21675" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.21675" title="Other formats" id="oth-2503.21675" aria-labelledby="oth-2503.21675">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Reducing of system of partial differential equations and generalized symmetry of ordinary differential equations </div> <div class='list-authors'><a href="https://arxiv.org/search/nlin?searchtype=author&query=Tsyfra,+I+M">I. M. Tsyfra</a>, <a href="https://arxiv.org/search/nlin?searchtype=author&query=Sitko,+P">P. Sitko</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Preliminary version </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Exactly Solvable and Integrable Systems (nlin.SI)</span>; Mathematical Physics (math-ph); Analysis of PDEs (math.AP) </div> <p class='mathjax'> Symmetry reductions of systems of two nonlinear partial differential equations are studied. We find ansatzes reducing system of partial differential equations to system of ordinary differential equations. The method is applied to system related to Korteweg -- de Vries (KdV) equation, and reaction-diffusion equations. We have also shown the possibility of constructing solution to system of non-evolutionary equations, which contains one or two arbitrary functions. </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 8 of 8 entries)</h3> <dt> <a name='item21'>[21]</a> <a href ="/abs/2310.07451" title="Abstract" id="2310.07451"> arXiv:2310.07451 </a> (replaced) [<a href="/pdf/2310.07451" title="Download PDF" id="pdf-2310.07451" aria-labelledby="pdf-2310.07451">pdf</a>, <a href="https://arxiv.org/html/2310.07451v2" title="View HTML" id="html-2310.07451" aria-labelledby="html-2310.07451" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2310.07451" title="Other formats" id="oth-2310.07451" aria-labelledby="oth-2310.07451">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Variational stabilization of degenerate p-elasticae </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Miura,+T">Tatsuya Miura</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Yoshizawa,+K">Kensuke Yoshizawa</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 23 pages, 3 figures, final version </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> J. Lond. Math. Soc. (2) 111 (2025), no. 3, Paper No. e70096 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Differential Geometry (math.DG) </div> <p class='mathjax'> A new stabilization phenomenon induced by degenerate diffusion is discovered in the context of pinned planar $p$-elasticae. It was known that in the non-degenerate regime $p\in(1,2]$, including the classical case of Euler's elastica, there are no local minimizers other than unique global minimizers. Here we prove that, in stark contrast, in the degenerate regime $p\in(2,\infty)$ there emerge uncountably many local minimizers with diverging energy. </p> </div> </dd> <dt> <a name='item22'>[22]</a> <a href ="/abs/2408.15100" title="Abstract" id="2408.15100"> arXiv:2408.15100 </a> (replaced) [<a href="/pdf/2408.15100" title="Download PDF" id="pdf-2408.15100" aria-labelledby="pdf-2408.15100">pdf</a>, <a href="https://arxiv.org/html/2408.15100v2" title="View HTML" id="html-2408.15100" aria-labelledby="html-2408.15100" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2408.15100" title="Other formats" id="oth-2408.15100" aria-labelledby="oth-2408.15100">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Joseph,+K+D">Kayyunnapara Divya Joseph</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> Hyperbolic systems of the first and higher-order partial differential equations appear in many multiphysics problems. We will be dealing with a wave propagation problem in a piece-wise homogeneous medium. Mathematically, the problem is reduced to analyzing two systems of partial differential equations posed on two domains with a common boundary. The differential equations may not be satisfied on the boundary (or part of the boundary), but some interface conditions are satisfied. These interface conditions depend on a specific physical problem. We aim to prove the existence and regularity of the solution for the case of hyperbolic systems of first-order equations with different domains separated by a hyperplane, where we need to formulate the interface conditions. We do this for the initial value problem in 1D-space variable when the coefficient matrix has discontinuity on $m$ lines. More specifically, we find explicit solutions to the case when the coefficient matrix is piecewise constant with a discontinuity along $1$ line or $2$ lines. We also prove the existence of solution to the general initial value problem. We then formulate the weak solution of initial value problem for the corresponding symmetric hyperbolic system in $n $D-space variables with interface conditions, get the energy estimates for this system, and prove the existence of solution to the system. </p> </div> </dd> <dt> <a name='item23'>[23]</a> <a href ="/abs/2412.11089" title="Abstract" id="2412.11089"> arXiv:2412.11089 </a> (replaced) [<a href="/pdf/2412.11089" title="Download PDF" id="pdf-2412.11089" aria-labelledby="pdf-2412.11089">pdf</a>, <a href="https://arxiv.org/html/2412.11089v2" title="View HTML" id="html-2412.11089" aria-labelledby="html-2412.11089" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2412.11089" title="Other formats" id="oth-2412.11089" aria-labelledby="oth-2412.11089">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The Lagrange Problem from the Viewpoint of Toric Geometry </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Tang,+X">Xiuting Tang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Symplectic Geometry (math.SG) </div> <p class='mathjax'> In this paper, I mainly prove the following results. For every energy value below the minimum of the first, second and third critical value, each bounded component of the regularized energy hypersurface of the Lagrange problem under some ranges of the parameters in the Hamiltonian function arises as the boundary of a strictly monotone toric domain, which is dynamically convex as a corollary. For the Euler problem as a special case of the Lagrange problem, when the energy is less than the negative value of the sum of the two masses of the fixed centers, the bounded component around the first fixed center of the regularized energy hypersurface of the Euler problem with two fixed centers with one positive and one negative mass arises as the boundary of a convex toric domain. Together with the result of Gabriella Pinzari, when the energy is less than the critical value, the toric domain defined above is concave for the case that the second mass is nonnegative, convex for the case that the mass is nonpositive. </p> </div> </dd> <dt> <a name='item24'>[24]</a> <a href ="/abs/2503.09307" title="Abstract" id="2503.09307"> arXiv:2503.09307 </a> (replaced) [<a href="/pdf/2503.09307" title="Download PDF" id="pdf-2503.09307" aria-labelledby="pdf-2503.09307">pdf</a>, <a href="https://arxiv.org/html/2503.09307v2" title="View HTML" id="html-2503.09307" aria-labelledby="html-2503.09307" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.09307" title="Other formats" id="oth-2503.09307" aria-labelledby="oth-2503.09307">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Nonlocal equations with kernels of general order </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ok,+J">Jihoon Ok</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Song,+K">Kyeong Song</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 27 pages, updated introduction and references </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> We consider a broad class of nonlinear integro-differential equations with a kernel whose differentiability order is described by a general function $\phi$. This class includes not only the fractional $p$-Laplace equations, but also borderline cases when the fractional order approaches $1$. Under mild assumptions on $\phi$, we establish sharp Sobolev-Poincar茅 type inequalities for the associated Sobolev spaces, which are connected to a question raised by Brezis (Russian Math. Surveys 57:693--708, 2002). Using these inequalities, we prove H枚lder regularity and Harnack inequalities for weak solutions to such nonlocal equations. All the estimates in our results remain stable as the associated nonlocal energy functional approaches its local counterpart. </p> </div> </dd> <dt> <a name='item25'>[25]</a> <a href ="/abs/2503.18876" title="Abstract" id="2503.18876"> arXiv:2503.18876 </a> (replaced) [<a href="/pdf/2503.18876" title="Download PDF" id="pdf-2503.18876" aria-labelledby="pdf-2503.18876">pdf</a>, <a href="https://arxiv.org/html/2503.18876v2" title="View HTML" id="html-2503.18876" aria-labelledby="html-2503.18876" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.18876" title="Other formats" id="oth-2503.18876" aria-labelledby="oth-2503.18876">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Singularity formation for the 1D model of EMHD </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+C">Chao Wu</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span> </div> <p class='mathjax'> In this paper, we study the singularity formation phenomenon of the 1D model of Electron Magnetohydrodynamics (EMHD). we will construct a solution whose $C^3$-norm blows up in finite time. In the end, we will show that the solution is in $C^{\infty}(\mathbb{R}\backslash \{0\})\cap C^{3,s}(\mathbb{R})\cap H^3(\mathbb{R})$ and is not asymptotically self-similar. </p> </div> </dd> <dt> <a name='item26'>[26]</a> <a href ="/abs/2405.12181" title="Abstract" id="2405.12181"> arXiv:2405.12181 </a> (replaced) [<a href="/pdf/2405.12181" title="Download PDF" id="pdf-2405.12181" aria-labelledby="pdf-2405.12181">pdf</a>, <a href="https://arxiv.org/html/2405.12181v3" title="View HTML" id="html-2405.12181" aria-labelledby="html-2405.12181" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2405.12181" title="Other formats" id="oth-2405.12181" aria-labelledby="oth-2405.12181">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Regularization by rough Kraichnan noise for the generalised SQG equations </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bagnara,+M">Marco Bagnara</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Galeati,+L">Lucio Galeati</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Maurelli,+M">Mario Maurelli</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Added a new section on the well-posedness of the two-dimensional linear transport equation with random drift and the same rough Kraichnan noise </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Analysis of PDEs (math.AP) </div> <p class='mathjax'> We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $\beta\in (0,1)$, an active scalar model interpolating between SQG ($\beta=1$) and the 2D Euler equations ($\beta=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $\theta_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $\alpha$ of the noise and the singularity degree $\beta$ of the velocity field; in particular, we can cover any $\beta\in (0,1)$ for suitable $\alpha$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise. </p> </div> </dd> <dt> <a name='item27'>[27]</a> <a href ="/abs/2412.03445" title="Abstract" id="2412.03445"> arXiv:2412.03445 </a> (replaced) [<a href="/pdf/2412.03445" title="Download PDF" id="pdf-2412.03445" aria-labelledby="pdf-2412.03445">pdf</a>, <a href="https://arxiv.org/html/2412.03445v3" title="View HTML" id="html-2412.03445" aria-labelledby="html-2412.03445" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2412.03445" title="Other formats" id="oth-2412.03445" aria-labelledby="oth-2412.03445">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Characterizing the range of the complex Monge-Amp猫re operator </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Liu,+S">Songchen Liu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 11 pages, no figures. v1. comments are welcome. v3 The main result remains unchanged, and the length of the paper has been reduced </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Complex Variables (math.CV)</span>; Analysis of PDEs (math.AP); Differential Geometry (math.DG) </div> <p class='mathjax'> In this note, we solve the complex Monge-Amp猫re equation for measures with a pluripolar part in compact K盲hler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss the properties of the complex Monge-Amp猫re operator in some special cases. </p> </div> </dd> <dt> <a name='item28'>[28]</a> <a href ="/abs/2501.07552" title="Abstract" id="2501.07552"> arXiv:2501.07552 </a> (replaced) [<a href="/pdf/2501.07552" title="Download PDF" id="pdf-2501.07552" aria-labelledby="pdf-2501.07552">pdf</a>, <a href="https://arxiv.org/html/2501.07552v2" title="View HTML" id="html-2501.07552" aria-labelledby="html-2501.07552" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2501.07552" title="Other formats" id="oth-2501.07552" aria-labelledby="oth-2501.07552">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Spectral distribution of the free Jacobi process with equal rank projections </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Demni,+N">Nizar Demni</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Hamdi,+T">Tarek Hamdi</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Analysis of PDEs (math.AP); Complex Variables (math.CV); Functional Analysis (math.FA); Spectral Theory (math.SP) </div> <p class='mathjax'> The free Jacobi process is the radial part of the compression of the free unitary Brownian motion by two free orthogonal projections in a non commutative probability space. In this paper, we derive spectral properties of the free Jacobi process associated with projections having the same rank $\alpha \in (0,1)$. To start with, we determine the characteristic curves of the partial differential equation satisfied by the moment generating function of its spectral distribution. Doing so leads for any fixed time $t >0$ to an expression of this function in a neighborhood of the origin, therefore extends our previous results valid for $\alpha = 1/2$. Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the compositional inverse of the $\chi$-transform of the spectral distribution of the free unitary Brownian motion. In this respect, we study mapping properties of this deformation and use the saddle point method to prove that the compositional inverse of a $\alpha$-deformation of the $\chi$-transform of the free unitary Brownian motion is analytic in the open unit disc (for large enough time $t$). The last part of the paper is devoted to a dynamical version of a recent identity pointed out by T. Kunisky in \cite{Kun}. Actually, this identity relates the stationary distributions of the free Jacobi processes corresponding to the sets of parameters $(\alpha, \alpha)$ and $(1/2,\alpha)$ respectively and we explain how it follows from the Nica-Speicher semi-group. Our dynamical version then relates the partial differential equations of the Cauchy-Stieltjes transforms of the densities of the finite-time spectral distributions. It also raises the problem of whether a dynamical analogue of the Nica-Speicher semi-group exists when the compressing projection has rank $1/2$. </p> </div> </dd> </dl> <div class='paging'>Total of 28 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AP/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em"> <!-- Macro-Column 1 --> <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div> <!-- End Macro-Column 1 --> <!-- Macro-Column 2 --> <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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