Papers

In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ \Phi(x,t,\nabla u))=f, \text{ in }\Omega \times (0,T). \] Here \(\Omega\) is a bounded open set of \(R^N\) with the boundary \(\partial \Omega\) satisfying Lipschitz condition. The Carathéodory function \(\Phi\) is restricted by $|\Phi(x,t,s)|\le c(x,t)|s|^\gamma$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q := \Omega \times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(\Omega)$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates.

For smooth axisymmetric Navier–Stokes flow with swirl, set \(F=u^\theta/r\), \(\Gamma=ru^\theta\), \(G=\omega^\theta/r\), and \(U=u^r/r\). Weighted swirl energies are classical; we determine the sharp two-parameter coercivity spectrum of the exact power-weight family \(\int |F|^p r^\alpha\,dr\,dz\), \(p>1\), \(\alpha>1\). The full diffusion form has a positive gradient gap for \(\alpha<2p+1\), loses every uniform gap at \(\alpha=2p+1\), and is indefinite above it. Exactly at the same threshold the radial-strain term cancels, and the identity becomes the classical \(L^p\) circulation energy for \(\Gamma\). Thus stretching cancellation and Hardy criticality coincide within the power family. We also characterize \(r^3drdz\) by conservative \(F\)-transport, self-adjoint five-dimensional diffusion, and a flat \(G\)-source pairing, and obtain the optimal strain form-bound threshold within the Hardy-coercive powers.

We give an exact finite-time decomposition of vortex stretching by resolving the vorticity equation according to the size of the vorticity. For almost every threshold $a>0$, the stretching occurring where $|\omega|>a$ equals the change of the corresponding superlevel mass plus two nonnegative viscous terms: one measuring changes in vorticity direction and one measuring dissipation across the level surface $|\omega|=a$. Integrating this identity over all thresholds recovers the classical $L^p$ vorticity balance and, for $p=2$, the usual spacetime palinstrophy. At the scale-critical exponent $p=3/2$, the dissipation can be written through the nonlinear variable $Z=|\omega|^{-1/4}\omega$. The threshold formulation also gives a positive measure that can be used to assign overlapping high-vorticity episodes without double counting. Under additional compactness assumptions it leads, through empirical transition couplings supported on a closed physical transition relation, to a stationary Markov renormalization law; no deterministic return map is required. These are structural results; no unconditional three-dimensional regularity theorem is claimed.

In this paper, we prove global well-posedness with large initial data for the one-dimensional quasilinear wave equation $$ u_{tt}=c(u)^2u_{xx}, \qquad (t,x)\in (0,T)\times\R, $$ where \(c\) is a positive, bounded, monotonically increasing function with bounded derivative. This result gives a partial resolution of an open problem posed by Glassey, Hunter and Zheng on the global existence of smooth solutions to this equation for large initial data. Our proof is based on upper and lower estimates for the Riemann variables via a new comparison principle.

In this work, we investigate entropy solutions for a class of systems of nonlocal {balance laws in which the convective flux and the source involves terms where the state variable convolved with kernels} in both spatial and temporal variables. This formulation captures the dependence of the flux on the solution within its spatial neighborhood (spatial nonlocality) as well as on its past states (temporal nonlocality), thereby incorporating memory effects. The resulting systems are coupled through these nonlocal interactions. We establish stability estimates for entropy solutions with respect to perturbations in the flux, the spatial and temporal kernels, and the initial data for the corresponding initial value problems. Finally, we present numerical experiments to illustrate the theoretical results and to highlight the influence of memory and source terms on the solution dynamics.

We prove two synchronization theorems for the second-order (inertial) Winfree model of coupled oscillators. The first result is a pathwise oscillator-death theorem with explicit smallness thresholds on the natural frequencies, initial velocities, and inertia, scaling as $R_0^{3/2}$ in the initial order parameter $R_0$. The second result is a qualitative zero-inertia synchronization statement: under generic initial data, if the intrinsic and initial velocity spreads are small compared to $\kappa$ and the inertia $m$ is small, then the limiting order parameter can be made arbitrarily close to 2. The proof of the first result is organized around three mechanisms, namely inertial gradient flow and the Łojasiewicz theorem, an initial layer argument, and an order-parameter bootstrapping argument. The proof of the second result involves approximation to the first-order case via a quantitative Tikhonov theorem.

We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding solution. Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. We then identify two principal mechanisms governing higher-order regularity: singular behavior of the kernel at the origin and jump discontinuities of the kernel, or of its derivatives, at the horizon endpoints. Singularities at the origin lead to blow-up of certain derivatives of the solution at the source discontinuity, while jumps at the horizon generate cascades of derivative discontinuities at translated locations. These phenomena occur for kernels commonly used in peridynamic-type models. By contrast, compactly supported \(C^\infty\) kernels do not generate derivative blow-up or cascading losses of regularity, and in this case the source term and the solution have equivalent piecewise smooth regularity. Motivated by this analysis, we develop a semi-analytic spectral method for the accurate numerical treatment of discontinuous nonlocal problems. The method uses successive smoothing transformations and explicitly constructed correction functions to convert the original problem into an auxiliary problem with improved regularity. A spectral solver is then applied to the smoothed problem, and the approximation to the original solution is recovered by adding back the analytic corrections. Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.

A Pólya-Szegö inequality for the circular rearrangement is proven, under general assumptions. In addition, sufficient conditions are given, under which all the extremals of the inequality are symmetric.

We announce a rigorous theory of optimal transport for signed (real) measures on Rd in the separated-support regime: the supports of the positive and negative parts of measures are assumed spatially disjoint. Under this structural hypothesis, together with regularity and non-degeneracy conditions on the absolutely continuous, discrete, and fractal singular components of the underlying Jordan and Lebesgue decompositions, we obtain existence and uniqueness of an optimal transport map for a cost that distinguishes same-sign and opposite-sign transports via a positional penalty, a coupled Monge-Ampere system together with a double Legendre transform characterising the solution, and exact preservation of Hausdorff dimension on the fractal singular components, the latter via a novel adaptive regularisation kernel and a direct measure-theoretic argument requiring no continuity of the transport map. This note states the results and outlines the two main proof techniques, full proofs, uniform estimates, and technical lemmas appear in the complete version of this work. We further report, in full and exhaustive summary, a complete numerical validation of this theory in ambient dimension one, two, and three: an entropic Sinkhorn solver for the single combined transport problem underlying the existence proof, applied to an explicit test case in which inter-sign transport is forced by mass imbalance and calibrated to occur with a large, robust cost margin, recovers exactly the predicted four-region structure with zero measured crossing of fractal mass between signs, and an exact-assignment refinement confirms pointwise-exact preservation of Hausdorff dimension under transport, with box-counting estimates converging to the theoretical dimension as resolution increases. The case of overlapping supports and the fully general unbalanced setting are the subject of Part II, in preparation.

The Lamb-Chaplygin dipole is a traveling wave solution to the 2D incompressible Euler equation, whose orbital stability was established in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] assuming the odd symmetry in $x_2$ (O) and non-negativity in upper half-plane (N). This paper is devoted to further study of its stability in the following two aspects. Firstly, we prove the spectral stability of the linearized operator around the Lamb-Chaplygin dipole without conditions (O) or (N), based on the index theory established in [Lin-Zeng, 2022]. This excludes an instability mechanism by unstable eigenmodes, and provides rigorous evidence towards nonlinear stability in this general setting. Secondly, assuming (O) and (N), we refine the orbital stability results in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] quantitatively by proving a linear bound of the fluctuation and a uniform control of the moving velocity. Instead of using a variational approach, our proof relies on the construction of a new coercive Lyapunov functional with a delicate mixed structure: it is quadratic in the interior region, but linear in the exterior region.

We give simple new proofs of two well-known results for the Schrödinger operator: first, the Brunn–Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including $C^{1,1}$ connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary.

We use the Unified Transform Method (UTM) for the vector case to resolve an interface problem for the Dirac equation on two semi-infinite domains and two finite domains in the massless and massive cases, respectively. The UTM for the vector case is a variation of the UTM for the scalar case. The solutions obtained for an interface problem on two semi-infinite domains and two finite domains, respectively, in the massive case are convergent explicit integral representations.

We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\mcA\vp=-\Div(A\nabla \vp)$, where $A=\diag(1,r^\al)$ and $\al\in(0,1)$. We establish well-posedness in weighted Sobolev spaces and prove an $L^2$ trace estimate for the normal derivative on the nondegenerate side $r=1$. Using truncated geometries and Carleman weights adapted to the anisotropic degeneracy, we derive a large-time observability estimate with a lower-order interior remainder. We also identify a framework-level obstruction at the critical threshold $\al=1$: the weighted Dirichlet coercivity underlying the subcritical analysis loses uniformity and exhibits a logarithmic loss on truncated domains.

We give a tug-of-war interpretation of the regularized $p$-Laplacian $\operatorname{div}\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $\Omega \subset \mathbb{R}^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $\Delta_p w=0$ in $\mathbb{R}^n+1$. Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\mathbb{R}^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$.

We study one-dimensional viscoelastic phase transitions modeled by a Ginzburg–Landau energy with a non-convex cubic stress-strain law. Extending the isothermal model, we couple the momentum equation to a heat equation for the temperature field, giving a thermoelastic system with viscous, capillary, and thermal-diffusion terms. We prove global existence and uniqueness of classical smooth solutions for the Cauchy problem, using a traveling-wave decomposition, an exponential transformation of the mechanical perturbation, and coupled energy estimates at successive regularity levels. Under additional integrability and small-data assumptions, the temperature perturbation decays algebraically.

We introduce a two-parameter continuity path for the J-equation and use it to characterize the solvability of the J-equation for Kähler metrics with Poincaré type singularities along a divisor $D$, allowing simple normal crossings and self-intersections. On Kähler surfaces, we show that the classical subsolution condition in the smooth setting implies solvability in the Poincaré type setting for any smooth divisor $D$. As a consequence, if $X$ contains no curves of negative self-intersections and $K_X[D]$ is ample, then the K-energy is bounded from below on any Poincaré type Kähler class. In the smooth divisor case, we further analyze the asymptotic behavior of solutions near $D$, and show that existence of a Poincaré type solution implies existence of a solution to the J-equation on $D$.

In this paper, we investigate a system of parabolic partial differential equations with unknown-dependent coefficients that integrates two models: an anisotropic orientation-adaptive denoising process in image processing and a phase-field model of grain-boundary motion in materials science. In recent years, several studies have attempted to develop a unified framework for treating these two research areas by considering pseudo-parabolic systems obtained through the introduction of the energy-dissipation operator $ - \Delta \partial_t $. However, the mathematical models for image processing and grain-boundary motion are originally formulated as parabolic systems. Therefore, establishing a unified analytical framework for such parabolic models remains an open problem. In this paper, we address this open problem by introducing a notion of solution that reproduces energy dissipation in parabolic systems, which we call an energy dissipative solution. As the main result, we clarify conditions that guarantee the existence of such solutions. The results of this paper establish a unified analytical framework for parabolic models, which has remained unresolved, and provide a solid theoretical foundation for advanced problems spanning both image processing and materials science.

Product-to-sum identities for trigonometric functions play a fundamental role in function theory and numerous applications. In this spirit, we present convolution-to-sum identities for Mittag-Leffler type functions. Using a Laplace domain analysis of fractional operators, we identify a family of Mittag-Leffler type functions that encapsulates the eigenfunctions of Riemann-Liouville and Caputo fractional derivatives. We work with two closely-related parameterizations of this class, $R_{\alpha,v}$ and $P_{\alpha,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $\alpha$, the convolution can be reduced to a sum of two $P/R$ functions through a partial-fraction decomposition in the Laplace domain. Furthermore, $R$ and $P$ functions satisfy a generalization of Euler's identity, which expands the scope of the previous result to convolutions of $P/R$ functions whose orders $\alpha_1,\alpha_2$ are related by a rational factor. For $\frac{\alpha_1}{\alpha_2} = \frac{n}{m}$, the resulting sum has $n+m$ terms. The foundational results and methods developed here are illustrated by their application to forced subdiffusion and to a fractionally attenuated wave equation (the Caputo-Wismer-Kelvin, or the fractional Kelvin-Voigt model).

We study the stability of minimizers of weighted $p$-area functionals associated with prescribed $p$-mean curvature surfaces in the Heisenberg group. While existence and uniqueness results are well established, quantitative stability with respect to perturbations of the mean curvature $H$ remains largely unexplored in the nonzero-$H$ regime. Using a Rockafellar–Fenchel duality framework, we identify a unique underlying vector field associated with each minimizer and prove its stability under perturbations of $H$. This yields quantitative control of the direction field of the horizontal gradient. Building on this structure, we establish $L^1$ stability of admissible minimizers under natural geometric assumptions on level sets. In dimensions two and three, we also derive $W^{1,1}$ stability estimates under additional regularity and structural hypotheses, with explicit rates in terms of $\|H-\tilde H\|_{L^\infty}$. Our results provide the first quantitative stability theory for $p$-area minimizing graphs with prescribed nonzero $p$-mean curvature, even in the unweighted case. Numerical simulations are included to illustrate the robustness of the theoretical results.

We consider a strongly damped semilinear wave equation with initial data prescribed as $(\varrho\phi,\varrho h)$, where the profiles are fixed and only the amplitude $\varrho>0$ is allowed to vary. The question addressed here is how this rescaling affects a guaranteed lower bound for the maximal existence time. We show that the solution exists at least on a time interval of length comparable to $\varrho^{-(p-2)}$. The proof is based on the growth of a quadratic phase-space norm: after the source term is estimated by the relevant Sobolev embedding, the problem reduces to a scalar differential inequality. The constants produced in the argument are independent of $\varrho$, so the dependence on the initial amplitude remains explicit throughout.

We investigate a hierarchical control problem for a one-dimensional Stefan system with localized distributed controls. The setting combines a Stackelberg strategy with a Nash equilibrium among multiple followers, yielding a multi-objective free-boundary problem. The interaction between the hierarchical control and the moving interface results in a nonlinear optimality system, and we show that the original problem reduces to the null controllability of this optimality system. Under suitable geometric conditions on the control regions, we establish a local null controllability result. The proof relies on an observability inequality for a linearized system, obtained through Carleman estimates adapted to the presence of a moving boundary. These results constitute, to the best of our knowledge, the first treatment of a Stefan system within a Stackelberg-Nash framework.

We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

In this paper, we study the long time behavior of solutions of Klein-Gordon-Zakharov system. We show that there exists a solution with special characteristics, which we shall refer to as a dipole solution, that is, there exists a solution $\vec{u}$ such that $$\left\|\vec{u}(t)-\sum_{k=1}^{2}\vec{R}_{k}\right\|_{X} \to 0 \, \, \text{as}\, \, t\to \infty,$$ where $\vec{R}_{k}$ represents a solitary wave for each $k$, with a translation $z_k$ with respect to its position, satisfying that $$|z_1(t)-z_2(t)| \sim 2\log(t)\, \, \text{as} \, \, t\to \infty.$$ Our approach will initially focus on the spectral analysis of the Hamiltonian operator associated with our system. Subsequently, we aim to establish a coercivity estimate that will allow us to derive conditions ensuring the existence of our solution. It is important to note that, in this problem, our objective is to obtain approximate solutions by solving a final data problem. These approximate solutions will then be used, through uniform estimates and compactness results, to derive the desired conclusions via density arguments.

We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a correct functional setting and to establish the existence of its non-trivial solution.

An improvement of a Global (strong) Gagliardo-Nienberg inequality with a BMO term is established by replacing local derivatives by and fractional Laplacians. Local versions are also given.

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form “maximise $\int \psi(\Theta_m)-c\int m$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=\chi_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem “maximise $\int \psi(\Theta_m)$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=\chi_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau & Weiss and Chanillo, Kenig & To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.

We develop a quantitative mathematical theory that offers new perspectives on nonlinear harmonic generation in plasmonic structures arising from symmetry breaking. Focusing on second harmonic generation–the most fundamental process and the most extensively studied owing to its practical significance–we establish a theoretical framework that can be readily extended to higher-order harmonics. We investigate the plasmonic system in the static regime using a columnar nanowire with \(n\)-fold rotational symmetry (\(n \in \mathbb{N}\)) and construct a phenomenological model in which the second harmonic response originates from nonlinear sources confined to a selvedge region near the surface. By introducing a notion of symmetry degree grounded in group theory, we precisely quantify the second harmonic generation in terms of multipolar contributions. Our theory complements existing physical descriptions of this practically important phenomenon and provides a rigorous account of how nonlinear optical efficiency depends on shape, size, symmetry, and defects in plasmonic structures.

We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star limit of a corresponding scalar nonlocal conservation law. The flux function of the nonlocal conservation law depends on suitable spatial averages of the density. The proof is based on a reformulation on the Hamilton–Jacobi level: working with the primitives, we identify the limit via the stability properties of viscosity solutions; we then recover the entropy solution using the classical relation between Hamilton–Jacobi equations and scalar conservation laws. We further show that the approximation extends, after a suitable shift, to sign-changing initial data, and we prove a quantitative convergence estimate for convex fluxes in terms of the first moments of the nonlocal kernels. This result makes it possible to define entropy solutions for general fluxes using their nonlocal approximations, which satisfy the requirement for a finite speed of mass propagation, a key feature of hyperbolic conservation laws.

For each parameter $a>1$, the critical hyperbolic catenoid $\Sigma_a$ is a rotationally symmetric, free boundary minimal annulus in a geodesic ball $B^3(r(a))\subset\mathbb{H}^3$. The Morse index of $\Sigma_a$ is at least $4$ by Medvedev [7], who conjectures equality. In this paper we identify a new geometric and spectral phenomenon for the family $\{\Sigma_a\}_{a>1}$, which we call "parameter-criticality", and study its consequences for the Robin spectrum. Specifically, we prove two main results: (I) Parameter-criticality (Theorem 1.5). The boundary radius $r(a)$ is non-monotone on $(1,\infty)$: it satisfies $r'(1^+)<0$ and $r(a)=\frac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$ with $d_\infty=\log[\Gamma(1/4)/\Gamma(3/4)]-\frac{1}{2}\log(2\pi)$ (Theorem 1.4). Hence there exists a parameter-critical value $a^\sharp\in(1,\infty)$ with $r'(a^\sharp)=0$. (II) Robin nullity jump (Theorem 1.6). At every such $a^\sharp$, the Robin nullity of $\Sigma_{a^\sharp}$ satisfies $\text{nul}(L_{\Sigma_{a^\sharp}})\geq 3$, with an additional kernel element in mode $k=0$ generated by the parametric variation field $j_a=\langle\partial_a\Phi_a,\nu\rangle_L|_{a=a^\sharp}$, which we show is non-vanishing at the catenoid neck via the closed-form $j_a(0)=1/(2\sqrt{a^2-1})$. The argument requires the limit $r_0:=\lim_{a\to 1^+}r(a)$ characterized as the unique positive solution of the transcendental equation $\tanh(r_0)\,\tanh(2r_0/\sqrt{3})=\sqrt{3}/2$ (Theorem 1.3), giving a clean parametrization of the degeneration $\Sigma_a\to\Sigma_1$. The Robin nullity of $\Sigma_a$ in mode $|k|=1$ is shown to equal $2$ (Proposition 1.1); this extends to the hyperbolic setting the mode-by-mode Fourier decomposition technique of Devyver [2] for the Euclidean critical catenoid, and is used in the proof of (II) to identify the extra kernel as a mode-$k=0$ phenomenon.

In this article, we investigate the asymptotic profile of solutions to the Cauchy problem for a nonlinear beam equation with two variable coefficients in the subcritical nonlinear case. In contrast to our previous result [6], in which the asymptotic profile is governed by the linear heat kernel and the nonlinear effect is asymptotically negligible, the asymptotic profile in the present setting is described by a self-similar solution to the associated nonlinear parabolic equation (constructed in Brezis-Peletier-Terman [1]). The proof relies on delicate energy estimates in weighted spaces formulated in parabolic self-similar variables.