Papers

We consider the Cauchy problem $$u_t-\Delta u=|u|^p,\qquad (t,x)\in(0,T)\times\R^n, \quad u(0,x)=\eps u_0(x),\qquad x\in\R^n,$$ with small sign-changing initial data having zero total mass. We assume $$u_0\in L^1(\R^n)\cap L^\infty(\R^n),\quad |x|u_0\in L^1(\R^n),\quad \int_{\R^n}u_0(x)\,dx=0,$$ and, for the sharp subcritical upper bounds, that $\int_{\R^n}x\,u_0(x)\,dx\neq0.$ Let $T_\eps$ denote the maximal lifespan. We identify a new transition exponent $p_m=1+\frac1{n+1}$ inside the Fujita range $1<p\le p_F:=1+2/n$, and establish the sharp lifespan estimates $$T_\eps\asymp \begin{cases} \eps^{-\frac{2(p-1)}{2-(n+1)(p-1)}}, &1<p<p_m,\\[2mm] \eps^{-\frac2{n+1}} \bigl(\log\frac1\eps\bigr)^{-\frac2{n+2}}, &p=p_m,\\[2mm] \eps^{-p\left(\frac{1}{p-1}-\frac{n}{2}\right)^{-1}}, &p_m<p<p_F,\\[2mm] \exp\!\left(\eps^{-p(p-1)}\right), & p=p_F. \end{cases}$$ These estimates reveal a lifespan phenomenon that is different from the classical Lee–Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size \(O(\eps^p)\). Their competition produces the additional threshold \(p_m\), the logarithmic correction at $p=p_m$, and, at $p=p_F$, a substantially longer critical lifespan with exponent $p_F(p_F-1)$ instead of the classical Lee–Ni exponent $p_F-1$. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified $L^1$–$L^\infty$ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although $p_F$ remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.

We prove rigidity near the Euclidean metric for the anisotropic Calderón problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable Hölder norm and having the same Dirichlet-to-Neumann map agrees with it up to a diffeomorphism fixing the boundary pointwise. The result applies to general smooth anisotropic perturbations, without analytic or quasianalytic regularity assumptions, a prescribed conformal class, or a transversal product structure, and requires no convexity of the boundary. The proof combines harmonic coordinates with a Carleman estimate for compactly supported correctors to obtain a quadratic Fourier estimate on a frequency range determined by the perturbation. A frequency decomposition then yields rigidity.

We study the inverse problem of recovering initial data for fractional heat equations on bounded domains from observations taken on two types of possibly Lebesgue-null sets: the first consists of general thin sets whose Hausdorff dimension exceeds \(n-1\); the second consists of specially constructed sets of zero Hausdorff dimension, built from algebraic irrational points and rapidly accumulating sequences.By extending thin-set observability inequalities from the classical heat equation to the fractional setting and employing a unified spectral inequality with exponent \(\beta\in(0,s)\), we establish a quantitative observability estimate for every fractional exponent \(s>1/2\). For general thin sets one has \(\beta=1/2\); for the special zero-dimensional sets any \(\beta\in(1/2,s)\) is admissible, yet the short-time observability cost always remains of exponential type, with the rate governed by \(\beta\) relative to \(s\). Under an a priori smoothness assumption, we also prove a logarithmic stability estimate.We further design a regularised least-squares reconstruction algorithm and provide a conditional convergence analysis, showing that the reconstruction error is bounded by the sum of a spectral truncation term and a logarithmic term dictated by the continuous stability, and that the error decays logarithmically as the noise level tends to zero. Numerical simulations using very few observation points confirm the feasibility of the proposed approach.

We consider the nonlinear wave equation (NLW) with a superlinear pure power nonlinearity in the subconformal and conformal cases. Under some conditions on initial data, this equation is known to have solutions which blow up in finite time. Two questions are then relevant: (i) the classification of all possible blow-up behaviors; (ii) the construction of examples of blow-up solutions. As we will show, the situation is entirely settled in the one-dimensional case, where we fully solve the famous soliton resolution conjecture. Various extensions to higher dimensions and to perturbed versions of NLW are given, including the construction of a solution in 2-d, with a nearly pyramidal blow-up graph. Carrying out this program was made possible thanks to a synergy of techniques from the PDE theory, mathematical physics, and analysis, including ODE techniques, spectral theory, and energy methods. Throughout this presentation, we will insist on connections with the study of other types of PDEs, in particular in the parabolic case. Surprisingly enough, in spite of the difference between parabolic and hyperbolic equations at the linear level, the nonlinear nature brings in a strong unity-both in the results and in the methods-between these two important classes of PDEs.

We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig–Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.

We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on $\R^d$, $d\ge2$, in the endpoint space \[ \dot B^{-1}_{\infty,1}(\R^d)\times \dot B^{0}_{\infty,1}(\R^d). \] For every sufficiently small $\varepsilon>0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and \[ \norm[\dot B^{-1}_{\infty,1}]{u_0} +\norm[\dot B^{0}_{\infty,1}]{b_0}<\varepsilon, \] such that the corresponding classical solution satisfies \[ \norm[\dot B^{0}_{\infty,1}]{b(t_\varepsilon)}>\varepsilon^{-1} \] for some $0<t_\varepsilon<\varepsilon$. Consequently, no solution map agreeing with classical solutions can be continuous at the origin in the magnetic endpoint topology. The low–high paraproduct mechanism available when the dyadic summability exponent is greater than one gives no gain at the $\ell^1$ endpoint. We instead combine a resonant four-wave velocity interaction with magnetic stretching to produce uniformly sized contributions on a growing family of dyadic shells. Frequency-localized test functionals extract the endpoint lower bound, while uniform Lagrangian estimates in weighted Fourier $L^1$ spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.

In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.

This paper investigates nonlinear eigenvalue problems governed by a family of operators that combine two distinct diffusion mechanisms: a classical local diffusion, which accounts for short-range interactions, and a fractional nonlocal diffusion, which captures long-range effects. The analysis is carried out in the Heisenberg group, a non-Euclidean geometric setting in which motion is constrained to a distinguished set of horizontal directions. As an application of the obtained spectral theory, we establish multiplicity results for perturbed singular problems in both the purely nonlocal and mixed local–nonlocal settings. For the mixed case, a gradient convergence theorem plays a crucial role in deriving these multiplicity results. A key feature of our analysis is the treatment of variable singular exponents, allowing the singularity to vary throughout the domain. To the best of our knowledge, this is the first systematic study of nonlinear eigenvalue problems and singular problems with variable singularity exponents for mixed local-nonlocal operators in the Heisenberg group. The results therefore open a new direction in the spectral theory of subelliptic operators and provide a rigorous mathematical foundation for applications ranging from anomalous diffusion and nonlocal phase transitions to image analysis and control theory on nonholonomic systems.

In the present paper, we study the maximization problem of the $k$-Torsional rigidity under quermassintegral constraint, with particular emphasis on the role of the ball. Our main result shows a phenomenon which, as far as we know, has not previously been observed: the ball, rather than being an extremal shape, is a saddle point. More precisely, for suitable quermassintegral constraints, we show that the $k$-torsional rigidity admits both increasing and decreasing perturbations around the ball. The proof is based on a second-order analysis of the corresponding variational problem. To the best of our knowledge, this is the first use of a second-order shape derivative approach in the study of variational problems for $k$-Hessian operators. We also establish an analogous saddle-point behavior for a related curvature functional under a volume constraint, highlighting a structural distinction between the intermediate Hessian regime and the classical Laplace and Monge-Ampère cases.

The purpose of this paper is to extend an article by Hilhorst, Mimura and Schätzle [18] about the limit as the latent heat coefficient tends to zero of a two-phase Stefan problem arising in biology. We introduce a rather general additive noise white in time and colored in space, and search for the limit of the solution of the corresponding stochastic Stefan problem as the latent heat coefficient vanishes. We first prove the existence and uniqueness of the weak solution of this problem, and then study the limit of the solution as the latent heat coefficient tends to zero. Unlike in [18], our method of proof is based upon an error estimate between the solution of the Stefan problem with positive latent heat and that of the Stefan problem with zero latent heat, which seems to be novel even in the deterministic case when no noise is added.

We classify finite-energy positive solutions to the Yamabe-type equation on the 15-dimensional octonionic Heisenberg group, whose algebra is non-associative. Extremals for the Folland-Stein-Sobolev inequality on this group are explicitly described. This confirms the conjecture of Garofalo and Vassilev [Duke Math. J. 2001] on the $15$ dimensional octonionic Heisenberg group.

We study the far-field behavior of solutions of the one-phase Bernoulli free boundary problem in the exterior of a ball, and of entire solutions with a single compactly supported inhomogeneity of the free boundary condition, which we call a defect. For solutions which blow down to a half-plane solution (proper solutions) we establish an asymptotic expansion at infinity: in dimension $d \geq 3$ the free boundary height converges to a limit at rate $|x|^{2-d}$ with a capacity-type coefficient, while in dimension $d=2$ the expansion carries a logarithmic term. A significant novelty is that the expansions are quantitative and uniform over all the proper solutions.

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model and symmetric perturbations of symmetric inhomogeneous stationary states that are naturally associated with a pendulum Hamiltonian flow. We first show the existence of families of such stationary states having compact support inside the separatrix region, constant value near the origin, and satisfying a linear stability (Penrose) criterion. We then describe the nonlinear dynamics of small compactly supported perturbations with finite regularity of this class of equilibria. More precisely, we prove that for large (but finite) times, in a modulated action-angle coordinate system the solution remains close to a modulated stationary state and that the magnetization of the perturbation satisfes decay estimates driven by the linear dynamics.

We study a diffusive nonlocal interaction equation generated by the squared volumes of random $n$-simplices. The nonlinear drift depends on the solution only through its mean and covariance matrix, which yields a closed finite-dimensional covariance system and an explicit representation of the measure-valued solution as an affine pushforward followed by Gaussian convolution. We use this reduction to prove global well-posedness for arbitrary initial data in $\mathcal P_2(\mathbb R^d)$. We then classify the stationary states and show that they coincide with the minimizers of the associated free energy. When $1\leq n<d$, the equilibrium is a unique isotropic Gaussian up to translation. By contrast, at the critical dimension $n=d$, the energy fixes only the determinant of the covariance matrix, and, when $d\geq2$, a continuum of anisotropic Gaussian equilibria appears. Finally, we prove convergence of every solution to one of these equilibria in quadratic Wasserstein distance and obtain explicit exponential convergence rates.

In this paper we prove some one-dimensional symmetry results for stable solutions to quasilinear anisotropic elliptic equations in $\mathbb R^2$ and $\mathbb R^3$. In this regard, we give an extension of a formula obtained in \cite{SZ1,SZ2, FSV}. The techniques used are inspired by the works of \cite{Fhdr,FSV}.

We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish Hölder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying Lévy measures, without imposing any ellipticity condition on the nonlocal operator. The Hölder exponent is given explicitly in terms of the order $\sigma$ $\in$ (0, 2) of the operator and the growth exponent m > 1 of the Hamiltonian. In particular, our approach applies to arbitrary superlinear Hamiltonians, including the delicate regime 1 < m < $\sigma$ < 2, and yields an improved regularity exponent when $\sigma$ $\in$ (1, 2). Assuming in addition a weak ellipticity condition on the nonlocal operator, we prove that viscosity solutions are globally Lipschitz continuous. The proof combines the Hölder regularity supplied by the coercive Hamiltonian with the regularizing effect of the nonlocal diffusion through an Ishii-Lions argument, allowing us to treat Hamiltonians with arbitrary superlinear growth. Finally, we provide a counterexample showing that, in the absence of ellipticity, Lipschitz regularity may fail if the spatial dependence of the Lévy measures is merely Hölder continuous, thereby illustrating the sharpness of our continuity assumptions.

We prove an $O(t^{1/4})$ bound on the support radius of nonnegative planar Euler vorticity with compactly supported $L^1$ initial data, removing the logarithmic factors from the classical confinement estimates. The result holds in the symmetrized vorticity formulation. A factorization of the interaction kernel yields a quadratic convolution inequality for high-order moments. Retaining this convolution allows an elementary comparison for finite sums of moments to control the full support. We also prove that the squared support radius is Hölder continuous in time with exponent $1/2$.

In this paper, we study a system of second-order integral-partial differential equations with interconnected obstacles. A particular case, is the Hamilton-Jacobi-Bellman (HJB for short) system associated with the optimal switching problem in the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution of the system, which is unique in the class of continuous bounded functions. The Lévy measure $\lambda$(.) is of infinite activity. The main tool we use is the associated system of reflected backward stochastic differential equations with jumps and interconnected obstacles for which we also study existence and uniqueness of the solution.

We present a formalization in Lean 4 of Mouhot and Villani's theorem on nonlinear Landau damping in $\mathbb T^d$ in all Gevrey regularity indices $s > 1/3$ for small backgrounds.

We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation $(c^{-1}\partial_t-\Delta_g+q)u=0$ on a conformally transversally anisotropic manifold. The partial input–output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform $O(h^{1/2})$ concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.

We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.

We study the free-boundary Euler–Monge–Ampère system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge–Ampère equation coupled to a nonlinear boundary condition. For each fixed relaxation parameter, we establish local well-posedness for uniformly convex initial domains of class $C^{2,\alpha}$, densities in $C^{1,\alpha}$, and arbitrary initial velocities in $C^{2,\alpha}$. The solution retains these spatial regularities, and the force potential belongs to $C^{3,\alpha}$ up to the moving boundary. Within uniformly bounded data families, the solution depends continuously on the initial data at every lower Hölder exponent.

We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.

We consider the two-species Lotka-Volterra competition-diffusion system where one species has a small diffusion rate relative to the other species and has a small competition coefficient. By the geometric singular perturbation theory, we prove the existence of the wavefront connecting the coexistence state to the trivial state, with traveling speed greater than the minimal speed. For such wavefronts, we show that the profiles of both species are monotone. Finally, we use the geometric singular perturbation theory to estimate the associated Evans function, and give the related stability results.

Motivated by the problem of asymptotic stability for ground states of the Klein–Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic–quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic–quintic Klein–Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_\omega$ be the positive radial ground state of the three-dimensional cubic–quintic elliptic equation \[ -\Delta Q_\omega+\omega Q_\omega-Q_\omega^3+Q_\omega^5=0. \] We study two related, but logically distinct, spectral problems as $\omega$ approaches the endpoint $3/16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian \[ L_+=-\Delta+\omega-3Q_\omega^2+5Q_\omega^4. \] This operator is also the exact linearized spatial operator around the static state $Q_\omega$ for the real scalar cubic–quintic Klein–Gordon equation with mass parameter $\omega$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic–quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3/16-\omega)^{-1}$. %The rotating complex Klein–Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic–quintic models.

In this paper, we study singular positive solutions to the semilinear elliptic equation $$ - \Delta u = f(u) ~~~~~~ \textmd{in} ~ \Omega \setminus \Gamma, $$ where $\Omega \subset \R^n$ is a bounded or unbounded domain, and $\Gamma \subset \Omega$ is a singular closed set with zero Newtonian capacity. When $\Omega = \R^n$ and $\Gamma \subset \{ x_1 = 0 \}$, we establish the symmetry of singular solutions with respect to the hyperplane $\{ x_1 = 0 \}$ and their monotonicity in the $x_1$-direction. For the case where $\Omega \subset \R^n$ and $\Gamma$ is a smooth closed manifold of dimension $\leq n - 2$, we show the asymptotic symmetry of singular solutions with respect to the normal direction of $\Gamma$. These results significantly improve those of Chen-Lin (Duke Math. J. 1995; Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001), Li (Invent. Math. 1996) and Sciunzi (J. Math. Pures Appl. 2017). Unlike their proofs, we employ an improved version of the moving sphere method, which also simplifies the analysis and extends its applicability.

We study the semiclassical limit from the magnetic Hartree–Fock equation to the magnetic Vlasov–Poisson equation in three spatial dimensions in the presence of a nonuniform, purely space-dependent magnetic field. We first establish propagation of regularity for solutions of the magnetic Vlasov–Poisson equation under exponentially weighted assumptions. Building on this result and a gauge-covariant magnetic Weyl quantization framework, we then derive explicit convergence bounds for the semiclassical limit in semiclassical Schatten norms. In particular, the estimates are formulated in terms of the magnetic field rather than a particular choice of vector potential and are therefore gauge covariant. The resulting semiclassical convergence holds for a suitable class of initial data on any fixed finite time interval.

We develop a two-dimensional inviscid model for extensible membranes with continuous vortex shedding from both edges, coupled to a weak spectral Galerkin discretization of the membrane equation that permits direct computation at zero bending rigidity. Viewing the pressure jump as a flux for bound vorticity explains how membranes respond to concentrated fluid loading. For fixed-fixed membranes, compliance draws the body toward the leading-edge vortex, delaying its detachment and enhancing lift primarily during start-up; after periodic shedding develops, compliance mainly amplifies the force oscillations rather than mean lift. Vortex detachment can unload heavy membranes into compression, producing bending-regularized wrinkles. For $R_2>0$, the membrane equation remains well-posed independently of the sign of the tension, provided the membrane remains an immersion. Releasing the trailing edge creates an intrinsic degeneracy: the tension vanishes at the free endpoint, making the $R_2=0$ problem formally underdetermined, while the weak formulation selects its bounded-energy branch. This vanishing-tension endpoint traps pressure disturbances and drives rapid tip-snapping; the dominant frequency and effective wave number approach their zero-rigidity limits with an inverse-logarithmic correction arising from a singular Bessel branch. Finally, decreasing the stretching modulus destabilizes periodic flutter, and the repeated fluid-driven departure and elastic return suggest a homoclinic tangle underlying the transition from periodic to chaotic flutter.

The recent work \cite{HeWuXiaZhang2025} introduced a finite-data inverse nodal framework for regular one-dimensional Sturm-Liouville operators. Nevertheless, formulating a multidimensional inverse nodal theory for Schrödinger operators remains a challenging open problem. This paper addresses this gap by establishing a finite radial inverse nodal theory for the radial spectral branch of Schrödinger operators in balls. In contrast with the interval case, the nodal data are the radii of spherical nodal hypersurfaces of radial eigenfunctions. The exact radial reduction preserves this PDE nodal geometry and leads naturally to a singular weighted Sturm–Liouville problem with weight $w(r)=r^{n-1}$, rather than to a regular interval model. We minimize the PDE distance $\|Q-Q_0\|_{L^p(B_R)}$ , equivalently the weighted distance $\|q-q_0\|_{L^p_w(0,R)}$, among all radial potentials matching finitely many prescribed spherical nodal radii. For the associated nodal constraint sets, we prove spectral regularity, nodal differentiability, weak closedness, exact realization, and a finite-codimensional $C^1$ manifold structure. Crucially, without the variational selection imposed by \(q_0\), the prescribed finite nodal radii fail to determine the potential uniquely: the admissible potentials are invariant under constant shifts, and the finite nodal constraints yield locally infinite-dimensional level sets at regular points.

The Riemann-Hilbert problem for the focusing coupled nonlinear Schrödinger (CNLS) equation is formulated on the basis of the corresponding $3\times3$ matrix spectral problem. We remove the discrete spectrum of initial RHP with the aid of Darboux transformations. Based on the $\bar{\partial}$-steepest descent method, we establish the long-time asymptotic behavior of solutions to the CNLS equation for initial condition in the weighted Sobolev space. Compared to the improved nonlinear steepest descent method, we improve the error estimate up to order $\mathcal{O}\left(t^{-3/4}\right)$. Furthermore, we obtain the asymptotic stability of multi-soliton solutions for CNLS equation and analyze the law of multi-soliton collision in the view-point of Yang-Baxter map.