Papers

The existence of minimizer for stability of the fractional Sobolev inequality and the stability of Beckner conformal Log-Sobolev inequality (the endpoint case of fractional Sobolev inequality) have been established by König and Chen-Lu-Tang. The existence of minimizer for stability of the Log-Sobolev inequality is unsolved. In this paper, we solve this problem. For every $n\geq2$, we prove that the sharp $L^2$-stability constant of Beckner conformal log-Sobolev inequality on $\mathbb S^n$ is attained and is strictly less than $\frac{2\log2}{n}$. Moreover, the normalized stability constant of the fractional Sobolev inequality converges to the stability constant of the log-Sobolev inequality as $s\to0^+$. After Hersch normalization, every sequence of $L^{p_{s_k}}$-normalized minimizers for stability of the fractional Sobolev inequality, with $s_k\to0^+$, has a subsequence converging strongly in $L^2(\mathbb S^n)$ to an $L^2$-normalized minimizer of stability of the log-Sobolev inequality.

We study the normalized surface system of an irrotational incompressible fluid of infinite depth beneath an elastic Euler–Bernoulli beam with structural damping. For every fixed damping coefficient $\gamma>0$, we establish global well-posedness for arbitrary initial data in $H^2(\R)\times\bigl(\dot H^{1/2}(\R)\cap\dot H^1(\R)\bigr)$. The main novelty is that the admissible data include finite energy initial potentials for which neither $\psi_0$ nor $u_0=\psi_0+G(\eta_0)\psi_0$ has an $L^2$ representative. We also prove the exact energy dissipation identity at the regularity of the constructed solutions. The main analytical difficulties are the possible failure of $L^2$ integrability at low frequencies and the justification of energy differentiation at the solution regularity. To handle the low-frequency difficulty, we fix the low-frequency part of the initial potential and evolving an $L^2$ remainder. Homogeneous Dirichlet–Neumann estimates and a high-frequency Fourier symmetrizer yield local well-posedness with $H^{-1}$ forcing. Continuous Fréchet differentiability of the energy in these affine coordinates, together with a chain rule on $H^1\subset L^2\subset H^{-1}$, justifies the dissipation identity. The Energy identity and coercivity provide uniform bounds that yields global continuation tharough the local theory. Iterating the local uniqueness and stability estimates gives global uniqueness within the stated affine solution class and Lipschitz dependence on bounded sets over every finite time interval.

We study the regularity of the free boundary for a multi-phase obstacle-type problem associated to quadrature domains, which arises from the minimization of the energy $$J(U,F):= \int_{B_1} (\frac 1 2 |\nabla U|^2 + F \cdot U)dx$$ among all competitors $U=(u^1,\ldots, u^N)$ with non-negative components $u^i \geq 0$ with disjoint supports, and constant forcing terms $f^i>0.$ We focus on the geometry of the free boundary at points where at least three distinct components meet (triple junctions).

We study interior Hessian estimates for $k$-convex solutions of Hessian quotient equations $\frac{\sigma_k(D^2u)}{\sigma_\ell(D^2u)}=1$ with $2\leq k\leq n$, $0 \leq \ell < k$, and $k-\ell \in\{1,2\}$. We first adopt the unified quantitative separation-propagation method of Li–Wu [LW26b] to derive a doubling inequality, and then use the compactness argument of Shankar–Yuan [SY25] to obtain the estimates. This approach also applies to a class of general Hessian equations.

We propose a model for the one-dimensional isothermal gas flow through a valve in a pipe. The valve aims at keeping the outgoing flow in an assigned interval; moreover, its opening and closing depend on a further parameter. The model is realized by a system of two conservation laws with suitable conditions coupling the ingoing and the outgoing flow. We study the corresponding Riemann problem, where non-classical shocks show up. Although simple, the model allows to study two key oscillatory behaviors of the flow, which are due, respectively, to the chattering of the valve, especially seen in the numerical simulations, and to the stabilizing hysteresis induced by the valve operation. We characterize the states leading to chattering and show how they depend on the parameters of the model. Through several numerical simulations, we study the interplay between the two above phenomena.

We construct $T^2$-invariant Einstein metrics with positive Ricci curvature on #_m(S^2 x S^2) for every integer $m>1$. These provide infinite family of Einstein metrics on closed $4$-manifolds with $\lambda>0$, which are in addition not Hermitian. We further establish the uniqueness of the $T^2$-invariant Einstein metrics on $S^4$, CP^2, S^2 x S^2, and CP^2#\bar{CP}^2.

We prove the uniqueness of global solutions to the two dimensional barotropic compressible Navier–Stokes equations within the regularity class introduced by Danchin and Mucha [14], for general $C^1$ pressure laws. A crucial assumption is that both solutions have bounded divergence, locally integrable in time, which ensures that their densities remain bounded without excluding vacuum or requiring spatial regularity of the density.

We consider the two-dimensional isentropic compressible Euler equations for a polytropic gas, $p(\rho)=k\rho^{\gamma}$. The $R$–vacuum state–$R$ configuration—two rarefaction waves bounding a vacuum—occurs when the two constant states move apart with sufficiently large relative velocity and is essentially the only vacuum-forming Riemann pattern in the absence of external forces. Local-in-time rarefaction fronts have been constructed only away from vacuum, where the sound speed is bounded away from zero; their construction through the vacuum edge, and hence the structural stability of the $R$–vacuum state–$R$ configuration, has remained open. We establish this stability for $\gamma\in(1,1.3]\cup\{\frac{5}{3},\frac{7}{5}\}$: for sufficiently small perturbations, the Riemann problem admits a continuous entropy solution on a fixed time interval, with the domain partitioned by four characteristic surfaces and the solution in the three central regions belonging to $W^{2,\infty}$ in the self-similar coordinates $(t,x_{1}/t,x_{2})$, even though it may contain logarithmic singularities of the form $c^{\alpha}(\ln c)^{\beta}$ at the vacuum edge. The proof develops a geometric formulation directly on the first-order system, together with a direct analysis of the initial slice at the vacuum edge, where the formal high-order expansion fails.

This paper examines the stability analysis of two-dimensional incompressible fluids in $L^2_{uloc}$ spaces. Extending the classical stability framework for Euler equations in Yudovich-type settings, we establish the convergence of Navier-Stokes solutions toward Euler solutions as viscosity $\nu \to 0$, under the weakened assumptions $u^\nu_0 \in L^\infty$ and $\omega^\nu_0 \in L^\infty$ without imposing decay conditions at infinity. We first derive a priori estimates for the Navier-Stokes equations in $L^\infty$ spaces. Then, using harmonic analysis tools including Littlewood-Paley decomposition and Bony's paraproduct calculus, we prove the $L^\infty$-convergence of the velocity field and $L^2_{uloc}$-convergence of the vorticity as $\nu \to 0$.

We study Strichartz estimates for orthonormal families of initial data when time is integrated against an $\alpha$-dimensional Frostman measure $\mu$, $0<\alpha\le1$, with coefficients measured in $\ell^\beta$. For the fractional Schrödinger, wave and Klein–Gordon equations, we obtain orthogonality gains in the admissible region and necessary conditions on $\beta$. For the fractional Schrödinger equation the exponents are sharp up to an endpoint; the optimality of the threshold $\beta=q/2$ requires $\mu$ to be Ahlfors regular. For the Schrödinger equation in dimensions $d\ge2$, we prove a Lorentz endpoint estimate and, if $d>2\alpha$, the critical exponent $\beta=q/2$ at every pair $2\le q,r<\infty$ with \[\frac d2(\frac12-\frac1r)\ge\frac\alpha q, \quad r>\frac{dq}{d-\alpha};\] on $r=\infty$ we obtain a restricted weak-type estimate. Since the constants depend on $\mu$ only through its Frostman constant, Lebesgue measure yields global-in-time critical estimates in this range when $d\ge3$. The proofs combine Frostman fractional integration, a spectral near–far argument at the endpoint, and asymmetric Hilbert–Schmidt bounds with orthogonal frequency-block summation. We also obtain further estimates in the low-dimensional cases $d=1$ (fractional Schrödinger) and $d=2$ (wave and Klein–Gordon) with $\alpha>1/2$, and formulate the consequences for densities of freely evolving states.

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to\infty$ and $\epsilon\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa>0$, the eigenvalues of $-\Delta$ in $D_{\epsilon,N}$ homogenize to those of the Schrödinger operator $-\Delta+\kappa\rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

We study minimizers of the optimal Sobolev quotients for the critical embeddings $D^{1,p}(\mathbb R^N;\mathbb R^m)\hookrightarrow L^{p^*}(\mathbb R^N;\mathbb R^m)$ and $W^{1,p}(\Omega;\mathbb R^m)\hookrightarrow L^{p^*}(\Omega;\mathbb R^m)$. In the whole space, we classify all minimizers as $\boldsymbol u=\boldsymbol c\,U_{\delta,x_0}$, where $\boldsymbol c\neq\boldsymbol0$ and $U_{\delta,x_0}$ is an Aubin–Talenti bubble. On bounded domains and for $\lambda>0$, we equip $W^{1,p}(\Omega;\mathbb R^m)$ with $ \|\boldsymbol u\|_\lambda^p :=\int_\Omega \bigl(|D\boldsymbol u|^p+\lambda|\boldsymbol u|^p\bigr) $. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $ -\boldsymbol\Delta_p\boldsymbol u =|\boldsymbol u|^{p^*-2}\boldsymbol u\ \text{in }\Omega. $ For the associated quotient, we prove existence and global $C^1(\overline\Omega)$ regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.

We study the two-dimensional non-rotating primitive equations with anisotropic viscosity and no thermal diffusion. For doubly periodic perturbations of a hydrostatic equilibrium with an affine temperature profile, we establish local Hadamard well-posedness in Sobolev spaces under strict stratification. The initial horizontal velocity and temperature have sufficient regularity and vanishing vertical means, but need not be small. The estimates close at the same Sobolev order for these variables despite the horizontal derivative loss in the diagnostic vertical velocity. Stratification-weighted energy estimates and a horizontal compensation argument avoid an additional temperature derivative while retaining the full vertical dependence and the vertically homogeneous velocity component determined by periodic compatibility. A one-sided comparison places higher-order norms of the more regular solution only in terms involving lower-order differences. This structure allows the Bona–Smith method to yield continuous dependence in the original Sobolev topology. We obtain a unique maximal strong solution in the strictly stratified class, together with continuous dependence of the diagnostic vertical velocity and pressure in their natural Sobolev spaces and lower semicontinuity of the maximal lifespan.

In this paper, we study the following Schrödinger-Newton system with a vanishing magnetic potential and an electric potential \begin{equation}\label{0.1} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} (-i\nabla +A(x))^2u+V(x)u=\Phi u, \ & x\in\mathbb{R}^3,\\ -\Delta \Phi =|u|^2, \ & x\in\mathbb{R}^3, \end{array} \right. \end{equation} where $i$ is the imaginary unit, $A:\mathbb{R}^3\to \mathbb{R}^3$ is a continuous magnetic potential, and $V:\mathbb{R}^3\to \mathbb{R}$ is a continuous electric potential. First, we show that the ground state energy of system with $A(x)\equiv0$ and $V(x)\equiv V_{\infty}>0$ is isolated. Using this fact together with a global compactness lemma, a barycenter function, and Brouwer degree theory, we prove that under suitable conditions on $A$ and $V$, system admits either a ground state solution or a high energy solution. Interestingly, whether a ground state solution exists is significantly affected by the magnetic potential $A$ and the electric potential $V$. Furthermore, we provide sufficient conditions that guarantee the existence of a ground state solution, as well as conditions under which no ground state solution exists and a high energy solution appears instead.

We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots,u_N)\in H^1(\mathbb{R})^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, $1<p<3$ and $N\geq 2$ are arbitrary. For the cubic case where $p=2$ and $N\ge 2$, we establish the following structural results of the system $(S)$: (i) all nontrivial solutions can be classified completely; (ii) the linearized operator at any nontrivial solution is non-degenerate; (iii) exact $L^2$-mass formulas of all nontrivial solutions are derived in terms of distinct Lagrange multipliers $\mu_i$. As the applications of the above structural results, we first show that the system $(S)$ with $p=2$ does not have any orthonormal solution for any $N\ge2$. We then investigate a fermionic variational problem $J_N(p)$ with $N\ge2$, whose minimizers correspond to ground states of the system $(S)$ under orthonormal constraints. More precisely, we prove that there exists a sufficiently small $\delta_N>0$ such that $J_N(p)$ is attained for any $p\in(2-\delta_N,2)$, whereas $J_N(p)$ is not attained for any $p\in[2,2+\delta_N)$. These settle some conjectures of [Gontier–Lewin–Nazar, ARMA, 2021] and [Frank–Gontier–Lewin, CMP, 2021].

The dynamical formation of a complete Bose-Einstein condensate and the asymptotic purification of trapped bosons coupled to a quantized radiation field are rigorously established. By tracking the macroscopic one-particle density matrix $\kappa_T$ through a nonlinear resonance cascade, it is demonstrated that radiative relaxation irreversibly depletes all excited trap modes, even when the initial macroscopic state $\kappa_0$ is genuinely mixed. Here $\kappa_T$ is an average, with respect to a de Finetti measure $\mu_0$, over mean-field particle-field orbits, so that the mixedness of $\kappa_0$ reflects a statistical ensemble of orbits. Under the assumptions of a non-vanishing initial ground state component and strictly positive downward Fermi-Golden-Rule rates, strong trace-norm convergence to a pure condensate is proven, $$ \lim_{T\to\infty} \| \kappa_T-|\chi_0\rangle\langle\chi_0| \|_{\mathfrak{S}_1(\mathfrak{h})} = 0 \, . $$ Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically, $$ \lim_{T\to\infty}S[\kappa_T]=0 \, , $$ without any claim of monotonicity in $T$. Thus, any initially mixed, positive-entropy state ($0<S[\kappa_0]<\infty$) is purified to the ground state $|\chi_0\rangle\langle\chi_0|$. For the proof, we derive the Hartree hierarchy of the coupled system of bosons and photons from the finite-$N$ BBGKY hierarchy of the microscopic quantum many-body system, establishing trace-norm convergence in the mean-field limit as $N\to\infty$. The limits are taken successively; first $N\to\infty$, then the weak-coupling limit, and finally $T\to\infty$. Although the global particle-field evolution remains unitary, the macroscopic one-particle state converges to a pure state as energy and correlations are continuously radiated into the dispersive field.

We use microlocal tools to define distinguished incoming and outgoing vacuum states of the quantum theory of massive Klein–Gordon fields in non-trapping asymptotically flat spacetimes, including radiative perturbations of Minkowski space. The asymptotic vacuum states satisfy the Hadamard condition, and the incoming and outgoing particle representations are unitarily equivalent, or in other words the quantum S-matrix is well-defined. These results are based on a description of scattering of classical fields which keeps track of the asymptotic distinction between positive- and negative-frequency solutions of the Klein–Gordon equation. Our emphasis is on developing the theory without recourse to explicit asymptotics for classical waves.

The objective of this paper is to study the existence and uniqueness of positive ground state solutions of the following Schrödinger problem \[ -\Delta u=u\log u^2+\mu\abs{u}^{p-1}u\qquad\text{in }\mathbb {R}^N, \] where $N\ge2$, $1<p<2^*-1$ and $\mu>0.$ We prove that there exists a positive ground state solution for every $\mu>0$ and every such solution is smooth, radially symmetric with respect to some point, strictly decreasing in the radial variable. Moreover, as $\mu\downarrow0,$ the positive ground state solutions centered at the origin will converge, uniformly over the ground-state set, to $U_0(x)=\exp(\frac{(N-\abs{x}^2)}{2})$ in the logarithmic Orlicz–Sobolev space and in $C^{2,\alpha}(\R^N)$ for every $\alpha\in(0,1),$ respectively. Furthermore, for sufficiently small $\mu,$ we also prove that the positive ground state solutions are unique up to translations and the kernel space of the real scalar linearized operator can be spanned by the translation derivatives. The proof combines minimization on the Nehari manifold with uniform compactness and an exterior coefficient estimate that prevents normalized difference and kernel sequences from vanishing locally.

In this paper, we establish vanishing viscosity limit of the 2D stationary Navier-Stokes equations in an expanding annulus with a fixed width. The flow is subjected to a large perturbation of a rotation of rigid body at one boundary and stays at rest at another boundary. Due to this different boundary condition setting, the boundary layer effects are different near the interior and exterior boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Taylor-Couette flow $\left(Ar+B\frac{1}{r}\right)e_\theta$ for some suitable constant $A$ and $B$, which is determined by using the principle of the Prandtl-Batchelor theory. The perturbation of the tangential velocity becomes large as the annulus expands, which causes heavy difficulties when we show the convergence of the approximate solution. Variously weighted energy estimates up to third-order derivatives of the error solution are constructed to close the energy of the nonlinear system.

We study the $s$-Schrödinger map equation \[ \begin{cases} \partial_t u = -u \wedge (-\Delta)^s u \quad \text{on } \mathbb{R}^n \times \mathbb{R}\\ u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^2\\ u(x,0) = u_0. \end{cases} \] For $s \in (1/2,1)$ and $n \geq 5$ we prove that for small initial data $u_0$ in the critical Besov space one can obtain a global solution.

In this paper, we research an anisotropic elliptic equation in the entire Euclidean space R^{N} . Under the assumption that the nonlinearity of the equation has exponential or critical growth, we prove, by means of the mountain-pass theorem, that the equation admits a nontrivial positive weak solution. Moreover, applying the mountain-pass theorem and Ekeland's variational principle, we show that a perturbation of the equation possesses two distinct positive weak solutions.

In this paper, we consider the Cauchy problem for the three-dimensional (3D) compressible micropolar equations with density-dependent viscosities. By classical $L^p$ estimates to elliptic system, defining suitable energy functions, after the elaborate index analysis, we establish the global existence of strong solutions to the system with all viscosities depending density as long as the initial density is large enough. By product, We also prove the existence of global large strong solutions with only large initial shear viscosity $\mu_1(\rho)$ and angular viscosity $\mu_2(\rho)$, which consists with the research work \cite{ZZ2014} of 3D compressible Navier-Stokes equations with constant viscosities and improve the recent result of \cite{HLZ} in some extent where they requiring both initial shear viscosity and bulk viscosity to be large.

In this paper, we obtain an $\varepsilon$-regularity criterion at one scale for 3D Navier-Stokes equations using a coarse-to-refined estimate argument based on mean-fluctuation decomposition. Making use of this $\varepsilon$-regularity criterion, we improve the upper bound on the box dimension of the interior singular set to $15/13$.

We study the stability of the variable coefficient wave equation $u_{tt}-u_{xx}-(1+x^{2})u_{yy}+\chi_{\mathbb G}u_{t}=0$ on the rectangle $(-1,1)\times(0,\pi)$ with Dirichlet boundary conditions, where the viscous damping acts on a vertical strip $\mathbb G=(\eta-\delta,\eta+\delta)\times(0,\pi)$. We establish a precise relationship between the location of the damping region and the rate of energy decay. We prove that the associated semigroup is exponentially stable whenever the closure of the damping strip meets the line $x=0$, i.e., $|\eta|\leq\delta$, despite the existence of rays that never enter the damped region. When the damping region is strictly bounded away from $x=0$, the semigroup decays at the sharp rate $(\log(2+t))^{-1}$. The proofs combine separation of variables, one-dimensional resolvent estimates, a reflection identity for the harmonic oscillator, and Gaussian quasimodes. We then consider the spectral Galerkin approximation based on the Dirichlet eigenfunctions of the constant coefficient Laplacian and show that, although the sequence of semigroups resulting from the approximation converges to the continuous one, it fails to be uniformly exponentially stable.

We study the inverse problem for the fractional conductivity equation with uniformly elliptic conductivities $\gamma\in L^\infty(\mathbb R^n)$. For every $n\geq1$ and $0<s<1$, we prove that the local exterior Dirichlet-to-Neumann map on any nonempty open subset of the exterior of a bounded open domain determines $\gamma$ uniquely almost everywhere in $\mathbb R^n$. No continuity, Sobolev regularity, or prescribed behavior at infinity is required. The proof works directly with the conductivity bilinear form and avoids the fractional Liouville reduction. It combines exterior determination at Lebesgue points, equality of weighted solutions, and a unique continuation principle for the fractional Laplacian in the weighted class \[ L_s=\left\{ g\in L^1_{\mathrm{loc}}(\mathbb R^n): \int_{\mathbb R^n} \frac{|g(x)|}{1+|x|^{n+2s}}\,dx<\infty \right\}. \] Specifically, if $g\in L_s$ and both $g$ and $(-\Delta)^s g$ vanish in a nonempty open set, with the latter equality understood distributionally, then $g=0$ almost everywhere. This result requires neither local boundedness nor Sobolev regularity of $g$. Its proof uses the Caffarelli–Silvestre extension and boundary unique continuation for the associated degenerate elliptic equation.

In this paper, we study planar compressible cavity flows fast-decaying vorticity and establish a Schauder estimate in large scale. As an application, we obtain an a priori asymptotic growth rate estimate of the cavity, that $s^{\frac{1}{2}-\delta}\lesssim h(s)\lesssim s^{\frac{1}{2}+\delta}$. This is by far the most accurate asymptotic estimate for compressible cavity flows and is probably the first paper that takes vorticity into account in the asymptotic analysis.

We prove that every nonnegative source \(f\in L^1(\Omega)\) in the Monge–Ampère equation \[ \det D^2u=f\quad\text{in }\Omega, \] on a bounded uniformly convex \(C^{1,1}\) domain \(\Omega\subset\mathbb R^n\), \(n\geq2\), is uniquely determined by generalized boundary measurements of Alexandrov solutions. The measurement is the extremal boundary slope, which extends the outward normal derivative and may equal \(+\infty\). For each \(\omega\in\mathbb S^{n-1}\), it suffices to know these slopes for the cylindrical Dirichlet data \[ \frac{t_k^\omega}{2}|P_\omega x|^2,\qquad t_k^\omega\to\infty, \] along any sequence of amplitudes, where \(P_\omega\) projects onto \(\omega^\perp\). The boundary value and extremal slope determine the interior subgradient image up to a Lebesgue null set. An exact change-of-variables formula and a large-amplitude limit then recover the Euclidean X-ray transform of \(f\), yielding uniqueness without a positive lower bound on the source. For nonnegative \(L^\infty\) sources, the extremal slopes are bounded and coincide with the interior normal derivatives, giving a Dirichlet-to-interior-Neumann formulation. The argument also extends to \[ \vartheta(\nabla u)\det D^2u=f, \] where the known weight \(\vartheta\) is locally integrable, Borel measurable, and bounded below by a positive constant. Conditional on existence of the corresponding Alexandrov solutions, the same cylindrical boundary-slope data determine every nonnegative \(L^1\) source.

In this paper, we establish large deviations of invariant measures for fractional stochastic reaction-diffusion equations with polynomial reaction term driven by unbounded multiplicative noise on R^n. Under a dissipativity condition, we first prove the Freidlin-Wentzell large deviation principle of solution trajectories uniformly over all initial data in L^2(R^n). The key ingredient is a positive-time L^\infty-estimate for differences of controlled solutions which is independent of all initial data and hence yields uniform convergence of controlled stochastic solutions over the entire tphase space. Then using the uniform trajectory estimates, contraction and a second-moment estimate, we further prove the large deviations of invariant measures of the equation with a good quasipotential without introducing weighted spaces or establishing exponential tightness. The non-compactness of Sobolev embeddings in R^n is circumvented by uniform spatial tail-ends estimates. This is the first result on large deviations of invariant measures of the equation with multiplicative noise on unbounded domains.

Peter Lax's 1968 discovery that certain nonlinear wave equations possess operator pairs (Lax pairs) opened the theory of completely integrable systems, yielding exact solutions, infinite conservation hierarchies, and perfectly elastic soliton collisions. When a nonlinear dispersive equation is provably non-integrable (admitting no Lax pair and only finitely many conservation laws), what mathematical structure governs its dynamics? The K(m,m) compacton equations, u_t + (u^m)_x + (u^m)_{xxx} = 0 for m >= 2, provide a testing ground: they support traveling waves with compact support, yet fail every known integrability test. We prove universal algebraic formulas determining the shape, width, and speed of every K(m,m) compacton. We develop a constraint-deficit mechanism explaining how the five independent conservation laws constrain emergent amplitudes and suppress the fundamental Fourier mode of the collision debris, while higher radiation modes remain unconstrained. The mechanism's conclusions rest on explicit hypotheses supported by numerical evidence. A phase-plane analysis reveals the compacton as a singular limit of smooth solitary waves through simultaneous regularity loss and support compactification. These results delineate intermediate territory between integrable order and generic nonlinear dynamics, organized by scaling universality and by the constraint deficit of a finite conservation structure.

We consider the 1-D focusing cubic NLS with a smooth, spatially-dependent potential which decays rapidly as $x \rightarrow \pm \infty$. As is well known, under sufficient regularity conditions, solutions of the cubic NLS with such a spatially-dependent potential conserve both mass and energy. Additionally, in the absence of a potential, the 1-D focusing cubic NLS admits boosted soliton solutions. Furthermore, in the presence of a potential, the interaction of the potential with boosted soliton solutions to the free case is of active interest in the literature, particularly with respect to the dependence of this soliton-potential interaction on the velocity of the incident soliton. We adapt a finite-difference scheme to the setting of a smooth, rapidly-decaying spatially-dependent potential and show that this adaptation preserves the appropriate mass and energy invariants. Additionally, we use this algorithm to model soliton-potential interactions at a variety of incident soliton velocities.