Partial symmetry of least energy nodal solutions to some variational problems

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions. Supported by NATO grant PST.CLG.978736. Supported by DFG grant WE 2821/2-1. Supported by NATO grant SPT.CLG.978736.     

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation. II

We continue our study on the Cauchy problem for the two-dimensional Novikov–Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the more involved case of a positive energy parameter. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining new estimates for two different frequency regimes, extending our previous results for the negative energy case [18]. The low frequency regime, which our previous result was not able to treat, is studied in detail. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. Then we combine the linear estimates with a Fourier decomposition method and $X^{s,b}$ spaces to obtain local well-posedness of NV at positive energy in $H^s$, $s>\frac{1}{2}$. Our result implies, in particular, that at least for $s>\frac{1}{2}$, NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev–Petviashvili equations. As a complement to our LWP results, we also provide some new explicit solutions of NV at zero energy, generalizations of the lumps solutions, which exhibit new and nonstandard long time behavior. In particular, these solutions blow up in infinite time in $L^2$.     

Global renormalized solutions and Navier–Stokes limit of the Boltzmann equation with incoming boundary condition for long range interaction

For the long range interaction, we prove the global existence of renormalized solutions to the Boltzmann equation with incoming boundary condition. Furthermore, as Knudsen number ? goes to zero, the limit to the incompressible Navier–Stokes limit with homogeneous Dirichlet boundary condition is justified when the boundary data of the scaled Boltzmann equation is close to the Maxwellian with order $O({?}^3)$ in the sense of boundary relative entropy.     

Stable blowup for the cubic wave equation in higher dimensions

We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.     

Bordered surfaces in the 3-sphere with maximum symmetry

We consider orientation-preserving actions of finite groups G on pairs $(S^3,\mathrm{\Sigma })$, where Σ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus $g>1$ the maximum order of such a G for all embeddings of a surface of genus g, and classified the corresponding embeddings.In the present paper we obtain analogous results for the case of bordered surfaces Σ (i.e. with non-empty boundary, orientable or not). Now the genus g gets replaced by the algebraic genus α of Σ (the rank of its free fundamental group); for each $\alpha >1$ we determine the maximum order $m_{\alpha }$ of an action of G, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into $S^3$. For example, the maximal possibility $12(\alpha ?1)$ is obtained for the finitely many values $\alpha =2,3,4,5,9,11,25,97,121$ and 241.     

Fast implicit schemes for the Fokker–Planck–Landau equation

We propose new implicit schemes to solve the homogeneous and isotropic Fokker–Planck–Landau equation. These schemes have conservation and entropy properties. Moreover, they allow for large time steps (of the order of the physical relaxation time), contrary to usual explicit schemes. We use in particular fast linear Krylov solvers like the GMRES method. These schemes allow an important gain in terms of CPU time, with the same accuracy as explicit schemes. This work is a first step to the development of fast implicit schemes to solve more realistic kinetic models. To cite this article: M. Lemou, L. Mieussens, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.     

Well-posedness of the Prandtl equation with monotonicity in Sobolev spaces

By using the paralinearization technique, we prove the well-posedness of the Prandtl equation for monotonic data in anisotropic Sobolev space with exponential weight and low regularity. The proof is very elementary, thus is expected to provide a new possible way for the zero-viscosity limit problem of the Navier–Stokes equations with the non-slip boundary condition.     

Local existence,lower mass bounds,and a new continuation criterion for the Landau equation

We consider the spatially inhomogeneous Landau equation with soft potentials. First, we establish the short-time existence of solutions, assuming the initial data has sufficient decay in the velocity variable and regularity (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass throughout the domain. The resulting lower bounds are sub-Gaussian, which we show is optimal. The proof of mass-spreading is based on a stochastic process, and makes essential use of nonlocality. By combining this theorem with prior results, we derive two important applications: $C^{\infty }$-smoothing, even for initial data with vacuum regions, and a continuation criterion (the solution can be extended as long as the mass and energy densities stay bounded from above). This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.     

Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations

This paper is concerned with the following fourth-order elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \Delta ^{2}u-\Delta u+V(x)u=|u|^{p-1}u,\,\mathrm{in}\,\mathbb {R}^{N},\\ u\in H^{2}\left( \mathbb {R}^{N}\right) , \end{array} \right. \end{aligned}$$

where \(p\in (2,\,2_{*}-1),\,u{\text {:}}\,\mathbb {R}^{N}\rightarrow \mathbb R.\) Under some appropriate assumptions on potential V(x),  the existence of nontrivial solutions and the least energy nodal solution are obtained by using variational methods.

     

Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball

We prove the existence and multiplicity of positive radial solutions to the nonlinear system

$\{\begin{array}{c}\hfill ?\mathrm{\Delta }{u}_i=\lambda K_i(|x|)f_i(u_j)\text{ in }\mathrm{\Omega },\hfill \\ \hfill d_i\frac{?u_i}{?n}+{\stackrel{?}{c}}_i(u_i)u_i=0\text{ on }|x|=r_0,\hfill \\ \hfill u_i(x)\to 0\text{ as }|x|\to \infty ,\hfill \end{array}$

for a certain range of $\lambda >0$, where $i,j\in \{1,2\},i\ne j$, $\mathrm{\Omega }=\{x\in {\mathbb{R}}^N:|x|>r_0>0\}$, $N>2,d_i\ge 0$, $K_i:[r_0,\infty )\to (0,\infty )$, $\stackrel{?}{c}:[0,\infty )\to [0,\infty ),f_i:(0,\infty )\to \mathbb{R}$ are continuous with possible singularity ±∞ at 0 and satisfy a combined superlinear condition at ∞.     

On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances

We consider the dispersive Degasperis–Procesi equation $u_t?u_{xxt}?\mathtt{c}{u}_{xxx}+4\mathtt{c}{u}_x?u{u}_{xxx}?3u_x{u}_{xx}+4u{u}_x=0$ with $\mathtt{c}\in \mathbb{R}?\{0\}$. In [15] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space $H^s$ with $s\ge 2$, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis–Procesi at any order is action-preserving.     

Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

We consider concentrated vorticities for the Euler equation on a smooth domain $\mathrm{\Omega }?{\mathbf{R}}^2$ in the form of

$\omega =\sum _{j=1}^N{\omega }_j{\chi }_{{\mathrm{\Omega }}_j},|{\mathrm{\Omega }}_j|=\pi r_j^2,\underset{{\mathrm{\Omega }}_j}{\int }{\omega }_{j}d\mu ={\mu }_j\ne 0,$

supported on well-separated vortical domains ${\mathrm{\Omega }}_j$, $j=1,\dots ,N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with ${\mathrm{\Omega }}_j$ being part of unknowns. For any given vorticities ${\mu }_1,\dots ,{\mu }_N$ and small $r_1,\dots ,r_N\in {\mathbf{R}}^{+}$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff–Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $?{\mathrm{\Omega }}_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the 2N-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2N-codim directions corresponding to the vortical domain shapes.     

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of

(1)$?\mathrm{\Delta }u=|\mathrm{?}u{|}^p\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a small $C^2$ perturbation of the unit ball in ${\mathbb{R}}^N$. For $\frac{N}{N?1}<p<2$ we prove that if Ω is a sufficiently small $C^2$ perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of $p>2$ we prove a similar result but now the positive weak solution u is contained in $C^{0,\frac{p?2}{p?1}}(\stackrel{￣}{\mathrm{\Omega }})$ and yet is not in $C^{0,\frac{p?2}{p?1}+\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ for any $\epsilon >0$.     

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of

$\{\begin{array}{ccc}?\mathrm{\Delta }u(y)\hfill & \hfill =\hfill & u{(y)}^p\text{ in }y\in {\mathrm{\Omega }}_t,\hfill \\ u\hfill & \hfill =\hfill & 0\text{ on }y\in ?{\mathrm{\Omega }}_t,\hfill \end{array}$

where ${\mathrm{\Omega }}_t$ is a sufficiently small $C^{2,\alpha }$ perturbation of the cone $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^N:x=r\theta ,r>0,\theta \in S\}$ where $S?S^{N?1}$ has a smooth nonempty boundary and where $p>1$ satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.     

Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions

We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.