Daomin Cao

• Professor
• Professor at Academy of Mathematics and Systems Science, Chinese Academy of Sciences

In this paper, we establish the uniqueness and nonlinear stability of concentrated symmetric traveling vortex patch-pairs for the 2D Euler equation. We also prove the uniqueness of concentrated rotating polygons as well. The proofs are achieved by a combination of the local Pohozaev identity, a detailed description of asymptotic behaviors of the so...

In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength an...

Helical Kelvin waves were conjectured to exist for the 3D Euler equations in Lucas and Dritschel \cite{LucDri} (as well as in \cite{Chu}) by studying dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these $m$-fold symmetric helical Kel...

This paper concerns the bubbling phenomena for the ‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exh...

In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere. Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stabl...

In this paper we present some classification results for the steady Euler equations in two-dimensional exterior domains with free boundaries. We prove that, in an exterior domain, if a steady Euler flow devoid of interior stagnation points adheres to slip boundary conditions and maintains a constant norm on the boundary, along with certain addition...

In this paper, we study desingularization of steady solutions of 3D incompressible Euler equation with helical symmetry in a general helical domain. We construct a family of steady helical Euler flows, such that the associated vorticities tend asymptotically to a helical vortex filament. The solutions are obtained by solving a semilinear elliptic p...

For the incompressible Euler flows in planar bounded domains $\Omega$, we prove that the vortex concentrating at the harmonic center of $\Omega$ causes the existence of vortexes with small vorticities near the boundary. These results are very different from the known results for the multi-vortex flows in which all the local vorticities are of the s...

We are concerned with the existence of periodic traveling-wave solutions for the generalized surface quasi-geostrophic equation (including incompressible Euler equation), also known as von Kármán vortex street. These solutions are of C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}...

Rossby-Haurwitz (RH) waves are important explicit solutions of the incompressible Euler equation on a two-dimensional rotating sphere. In this paper, we prove the orbital stability of degree-2 RH waves in $L^p$ norm of the absolute vorticity for any $1<p<+\infty$, which verifies a conjecture of A. Constantin and P. Germain in [Arch. Ration. Mech. A...

By studying the contour dynamics equation and using the implicit function theorem, we prove the existence of stationary vortex patches with fixed vorticity and total flux for each patch for the surface quasi-geostrophic equation in a bounded domain near non-degenerate critical points of the Kirchhoff-Routh function.

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff–Rott operator, and accomplished by using implicit function theorem at point v...

We develop a new structure of the Green’s function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation f...

In this paper, we consider the problem of Schr\"odinger equation coupled with a neutral scalar field. By constructing solutions with multiple peaks, we prove that the number of non-radial solutions of this problem goes to infinity as Maxwell coupling constant tends to infinity. Chern-Simons limit of those solutions are also discussed.

We are concerned with the focusing \(L^2\)-critical nonlinear Schrödinger equations in \({{\mathbb {R}}}^d\) for dimensions \(d=1,2\). The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of K pseudo-conformal blow-up solutions particularly with the low rate \((T-t)^{0+}\), as \(t\rightarrow T...

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we s...

In this paper, we study nonlinear orbital stability of steady vortex rings without swirl, which are special global solutions of the three-dimensional incompressible Euler equations. We prove the existence of orbitally stable steady vortex rings. The proof is based on the classical variational method.

We develop a new structure of the Green's function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation f...

In this paper, we are concerned with the uniqueness and nonlinear stability of vortex rings for the 3D Euler equation. By utilizing Arnold 's variational principle for steady states of Euler equations and concentrated compactness method introduced by P. L. Lions, we first establish a general stability criteria for vortex rings in rearrangement clas...

In this paper, we are concerned with nonlinear desingularization of steady vortex rings in R 3 with a general nonlinearity f . Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on f are very general, an...

We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov–Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function r...

In this paper, we construct a family of traveling wave vortex pairs with specific forms like Lamb Dipoles for the quasi-geostrophic shallow-water(QGSW) equations. The solutions are obtained by maximization of a penalized energy with multiple constraints. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variat...

We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function ψε satisfies{−ε2Δψε=∑i=1k1Bδ(z0,i)(ψε−με,i)+γ,inΩ,ψε=0,onΩ, with ε→0+ the scale parameter of vortices, γ∈(0,∞), Ω⊂R2 a bounded simply connected Lipschitz domain, z0,i∈Ω the limiting location of ith vortex, and με,i t...

In this paper, we study desingularization of steady solutions of 3D incompressible Euler equation with helical symmetry in a general helical domain. We construct a family of steady Euler flows with helical symmetry, such that the associated vorticities tend asymptotically to a helical vortex filament. The solutions are obtained by solving a semilin...

In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form...

We construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we const...

In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and traveling-wave solution...

In this paper, we study the existence and asymptotic properties of the traveling vortex pairs for the two-dimensional inviscid incompressible Boussinesq equations. We construct a family of traveling vorticity pairs, which constitutes the de-singularization of a pair of point vortices with equal intensity but opposite sign. Using the improved vortic...

In this paper, we construct smooth traveling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the 2D incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible...

This paper is devoted to the study of steady vortex rings in an ideal fluid of uniform density, which are special global solutions of the three-dimensional incompressible Euler equation. We systematically establish the existence, uniqueness and nonlinear stability of steady vortex rings of small cross-section for which the potential vorticity is co...

In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equation \begin{document}$ i\partial_{t}\psi = -\Delta \psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast |\psi|^2)\psi- \lambda_3|\psi|^p \psi,\; \; \; (t,x)\in \mathbb{R}^+\times \mathbb{R}^3, $\end{document} where \begin{d...

We study the existence of different vortex-wave systems for inviscid gSQG flow, where the total circulation are produced by point vortices and vortices with compact support. To overcome several difficulties caused by the singular formulation and infinite kinetic energy, we introduce a modified reduction method. Several asymptotic properties of the...

In this paper, we constructed a family of steady vortex solutions for the lake equations with a general vorticity function, which constitutes a desingularization of a singular vortex. The precise localization of the asymptotic singular vortex is shown to be the deepest position of the lake. We also study global nonlinear stability for these solutio...

By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and traveling-wave vortex solutions for the gSQG equation, which extends the result of Hmidi and Mateu [28] to α∈[1,2). Moreover, we obtain the C∞ regularity of vortices boundary and the convexity of each vortices co...

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of traveling vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal intensities but opposite signs. The re...

In this chapter we are concerned with the non-steady Navier-Stokes and Stokes problems corresponding to the steady problems in Chap. 5. In Sect. 6.1 relying on the results of Sect. 3.1, we embed all boundary conditions to variational formulations. We get variational inequalities with one unknown which are equivalent to the original PDE problems for...

In this chapter we are concerned with the non-steady Navier-Stokes equations and Stokes equations with mixed boundary conditions including conditions for velocity, pressure, stress, vorticity and Navier slip condition together. As in Sect. 3.2, relying on the result in Sect. 3.1, we embed all these boundary conditions into variational formulations...

In this chapter, we outline some knowledge of analysis: Banach space, fixed point, Lebesgue and Sobolev spaces, operator and operator-differential equations and convex functional, which will be used in the main part of this book. We do not describe the best results, but to help readers’ understanding sometimes we say more than necessary. The reader...

In this chapter, we first show how the Navier-Stokes equations and the equations of motion for fluid under consideration of heat are derived. Next, we outline some boundary conditions for the Navier-Stokes equations, mainly being concerned with the ones dealt with in this book. Last, we consider three kind of bilinear forms for the Stokes and Navie...

In this chapter we are concerned with the equation for steady flow of heat-conducting incompressible Newtonian fluids with dissipative heating under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for te...

In this chapter we are concerned with a non-steady system for motion of incompressible Newtonian heat-conducting fluids with mixed boundary conditions. The boundary condition for fluid is the case of total pressure and the boundary conditions for temperature may include Dirichlet, Neumann and Robin conditions together.

In this chapter we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. We wi...

In this chapter we are concerned with the steady Navier-Stokes systems with mixed boundary conditions which may include Tresca slip condition, leak boundary condition, one-sided leak boundary conditions, velocity, pressure, vorticity, stress and normal derivative of velocity together. Relying on the results in Sect. 3.1 and using the strain bilinea...

In this chapter, we are concerned with the steady Navier-Stokes systems with mixed boundary conditions involving Dirichlet, pressure, vorticity, stress and normal derivative of velocity together. As we have seen in Sect. 2.3.2, according to what kinds of bilinear forms for variational formulation are used, types of boundary conditions under conside...

In this chapter we are concerned with the non-steady Boussinesq problem corresponding to the steady problem in Chap. 7. The formulations consist of a non-steady variational inequality for velocity and a non-steady variational equation for temperature. For the problem with boundary conditions involving the static pressure and stress, it is proved th...

We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $\Omega\subset \mathbb{R}^2$. Let $\kappa_i\not=0$, $i=1,\ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $\mathbf{x}_0=(x_{0,1},\ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $\Omega^m$ corresponding to $(\...

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point v...

In this paper, we prove nonlinear stability of planar vortex patches concentrated near a non-degenerate minimum point of the Robin function in a general bounded domain. These vortex patches are stationary solutions of the two-dimensional incompressible Euler equations. The result is obtained by showing that these concentrated vortex patches are in...

We are concerned with the focusing $L^2$-critical nonlinear Schr\"odinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $t\to T$, $1\leq K<\infty$. Moreover, we...

We study the existence of $x_2$-directional periodic traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation, known as von K\'arm\'an vortex street. These solutions are of $C^1$ type, and obtained by studying a semiliner problem on typical period. By a variational characterization to solutions, we also show the relati...

In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions for to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and travelling-wave sol...

By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed...

In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes...

In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with $N$-fold symmetry. The traveling-...

By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu \cite{HM} to $\alpha\in[1,2)$. Moreover, we prove the $C^\infty$ regularity of vortices boundary, and show the convex...

In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation−Δu=λu+μ(|x|−α⁎|u|2)u+(|x|−β⁎|u|2)u,x∈RN under the normalized constraint∫RNu2=c>0, where N≥3, μ>0,...

In this paper, we construct a family of symmetric vortex patches for the 2D steady incompressible Euler equations in a disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches “shrink” to...

This introduction to the singularly perturbed methods in the nonlinear elliptic partial differential equations emphasises the existence and local uniqueness of solutions exhibiting concentration property. The authors avoid using sophisticated estimates and explain the main techniques by thoroughly investigating two relatively simple but typical non...

In this paper we are concerned with the well-known Brezis-Nirenberg problem { − Δ u = u N + 2 N − 2 + ε u , a m p ; in Ω , u > 0 , a m p ; in Ω , u = 0 , a m p ; on ∂ Ω . \begin{equation*} \begin {cases} -\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, &{\text {in}~\Omega },\\ u>0, &{\text {in}~\Omega },\\ u=0, &{\text {on}~\partial \Omega }. \end{ca...

In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we cons...

This monograph explores the motion of incompressible fluids by presenting and incorporating various boundary conditions possible for real phenomena. The authors’ approach carefully walks readers through the development of fluid equations at the cutting edge of research, and the applications of a variety of boundary conditions to real-world problems...

In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any \begin{document}$ n $\end{document}-dimensional spaces and establish some...

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results...

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variat...

In this paper, we study the Cauchy problem of the nonlinear Schr\"{o}dinger equation with a nontrival potential $V(x)$. Especially, we consider the case where the initial data is close to k solitons with prescribed phase and location, and investigate the evolution of the Schr\"{o}dinger system. We prove that over a large time interval, all k solito...

In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in $\mathbb{R}^3$. The construction is based on a study of solutions to the...

In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)*|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad \text{in} \,\,\, \mathbb{R}^{2}, \end{eqnarray*} where $a\in C(\mathbb{R}^{2})$ is a $\mathbb{Z}^{2}$-periodic function with $\inf_{\mathbb{R}^{2}}a>0$,...

\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schr\"{o}dinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation \begin{equation*}\label{1} -\Delta u=\lambda u+\mu (|x|^{-\alpha}\ast|u|^{2})...

We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function $\psi_\varepsilon$ satisfies \begin{equation*} \begin{cases} -\varepsilon^2\Delta \psi_\varepsilon=\sum\limits_{i=1}^k \mathbf1_{B_\delta(z_{0,i})}(\psi_\varepsilon-\mu_{\varepsilon,i})_+^\gamma,\ \ \ & \text{in} \ \...

In this paper, we construct two types of vortex patch equilibria for the two-dimensional Euler equations in a disc. The first type is called the “N+1 type” equilibrium, in which a central vortex patch is surrounded by N identical patches with opposite signs, and the other type is called the “2N type” equilibrium, in which the centers of N identical...

In this paper, we study the desingularization of steady lake model of perturbation type with general nonlinearity f. Using the modified vorticity method, we construct a family of steady solutions with vanishing circulation, which constitute a desingularization of a singular vortex. The localization of the singular vortex is determined only by the v...

Optical vortices are phase singularities nested in electromagnetic waves which constitute a fascinating source of phenomena in the physics of light and display deep similarities to their close relatives, quantized vortices in superfluids and Bose–Einstein condensates. The present paper is concerned with the existence of stationary optical vortices...

In this paper, we establish a decay estimate for the fractional and higher-order fractional H\'enon-Lane-Emden systems, which deduces a result of non-existence. We also derive a Liouville type theorem via the method of scaling spheres introduced in \cite{DQ2}.

We constructed a family of steady vortex solutions for the lake equations with general vorticity function, which constitute a desingularization of a singular vortex. The precise localization of the asymptotic singular vortex is shown to be the deepest position of the lake. We also study global nonlinear stability for these solutions. Some qualitati...

In this paper we are concerned with a non-steady system for motion of incompressible viscous Newtonian heat-conducting fluids under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions based on total stress, velocity, total pressure, rotation, total stress together and the conditio...

In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow, there exists a family of steady Euler flows with small circulation in which the vorticity is continuous and suppor...

In this paper, we are concerned with nonlinear desingularization of steady vortex rings in $\mathbb{R}^3$ with given general nonlinearity $f$. Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on $f$ ar...

In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and can be regarded as desingularization of point vortices. By an adaption of the vorticity method, we construct a...

In this paper, we investigate desingularization of steady vortex rings in three-dimensional axisymmetric incompressible Euler fluids with swirl. Using the variational method proposed by Turkington, we construct a two-parameter family of steady vortex rings, which constitute a desingularization of the classical circular vortex filament, in both infi...

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variat...

In this note, we give a general criterion for steady vortex flows in a planar bounded domain. More specifically, we show that if the stream function satisfies “locally” a semilinear elliptic equation with monotone or Lipschitz nonlinearity, then the flow must be steady.

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~\Omega},\\ u>0, &{\text{in}~\Omega},\\ u=0, &{\text{on}~\partial \Omega}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>...

In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function q corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of...