Generalized Homoclinic Solutions of a Coupled Schr?dinger System Under a Small Perturbation

This paper is devoted to study a coupled Schr?dinger system with a small perturbation $$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$ where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schr?dinger-KdV system.     

Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

We consider the following nonlinear Schrödinger system in ${\mathbb{R}^3}$ $$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$ where P(r) and Q(r) are positive radial potentials, ${\mu > 0, \nu > 0}$ and ${\beta \in \mathbb{R}}$ is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.     

A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.     

A connection formula for the second Painlevé transcendent

We consider the second Painlevé transcendent $$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$ It is known that if y(x) ～ k Ai (x) as x → + ∞, where ?1<k<1 and Ai (x) denotes Airy's function, then $$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$ where the constants d, c depend on k. This paper shows that $$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$ , which confirms a conjecture by Ablowitz & Segur.     

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

In this paper, we consider the generalized Navier?CStokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier?CStokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier?CStokes equations by the more general operator (???) ^?? with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier?CStokes equation is local well-posed for a large set of the initial data in H ^?1+, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .     

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k^*(α) such that when k=k^*(α) the condition (2) is also satisfied. If 0 ^*, the solution exists for all x and tends to zero as x→-∞, while if k>k ^* then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^*=1, confirming previous numerical estimates.     

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$\begin{aligned} u_t(t,x)=(\mathcal {A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in \mathcal {H}, \end{aligned}$$ where $\mathcal {H}=\mathbb {R}^N$ or $\mathbb {Z}^N,\; \mathcal {A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal {H}=\mathbb {R}^N$ and is a discrete dispersal operator in the case $\mathcal {H}=\mathbb {Z}^N$ , and $f$ is periodic in $t$ , asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to $0$ as $\Vert x\Vert \rightarrow \infty $ for some time and space periodic function $f_0(t,x,u)$ ), and is of KPP type in $u$ . It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

A KAM Theorem for Higher Dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrödinger equation $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(|u|^2)u=0, \quad t\in \mathbb{R }, x\in \mathbb{T }^d \end{aligned}$$ with periodic boundary conditions, where $M_\xi $ is a real Fourier multiplier and $f(|u|^2)$ is a real analytic function near $u=0$ with $f(0)=0$ . We obtain for the equation a Whitney smooth family of real-analytic small-amplitude linearly-stable quasi-periodic solutions with a nice linear normal form.     

Double Obstacle Problems with Obstacles Given by Non-C 2 Hamilton–Jacobi Equations

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton–Jacobi equations that are not C ². When the Hamilton–Jacobi equation is not C ² then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton–Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C ¹-solutions to the Hamilton–Jacobi equation $$\pm |\nabla h-a(x)|^2=\pm 1\,{\rm in}\,B_1,\quad h=f \,{\rm on}\, \partial B_1$$ , are, in fact, C ^1,α/2, provided that ${a \in C^\alpha}$ . This result is optimal and, to the authors’ best knowledge, new.     

Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier–Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R ³. We first obtain the higher order regularity estimates for the solutions to Prandtl’s equation boundary layers. Furthermore, we prove that the strong solution to Navier–Stokes equations converges to the Eulerian one in C([0, T]; H ¹(Ω)) and ${L^\infty((0,T) \times \Omega)}$ , where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.     

Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity

In this paper, we study real solutions of the nonlinear Helmholtz equation $$- \Delta u - k^2 u = f(x,u),\quad x\in \mathbb{R}^N$$ satisfying the asymptotic conditions $$u(x)=O\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm as}\, r=|x| \to\infty.$$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein–Gordon equations with arbitrarily large frequency.     

Maximal and Minimal Spreading Speeds for Reaction Diffusion Equations in Nonperiodic Slowly Varying Media

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is ${f(x,u) = \mu_0 (\phi (x)) u(1-u)}$ , where??? ₀ is a 1-periodic function and ${\phi}$ is a ${\mathcal{C}^1}$ increasing function that satisfies ${\lim_{x \to+\infty}\phi (x) = +\infty}$ and ${\lim_{x \to +\infty}\phi' (x) =0}$ . Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of ${\phi}$ at infinity. If ${\phi}$ grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If ${\phi}$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation w _??, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.     

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

Parabolic Systems with p, q-Growth: A Variational Approach

We consider the evolution problem associated with a convex integrand ${f : \mathbb{R}^{Nn}\to [0,\infty)}$ satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings ${u\colon \Omega_T \to \mathbb{R}^n}$ which solve $$ \partial_tu-{\rm div} Df(Du)=0 $$ weakly in ${\Omega_T}$ , variational solutions exist under a much weaker assumption on the gap q ? p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and $$\frac{2n}{n+2} < p \le q < p+1.$$ These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption $$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$ we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$      

Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights

Let A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ?? ^k . We consider the variational eigenvalue problem (P) $$\mathcal{A}u = \lambda r{\text{(}}x{\text{)}}u,{\text{ }}x \in \Omega ,$$ , with Dirichlet or Neumann boundary conditions; here the “weight” r is a real-valued function on Ω which is allowed to change sign in Ω or to be discontinuous. Such problems occur naturally in the study of many nonlinear elliptic equations. In an earlier work [Trans. Amer. Math. Soc. 295 (1986), pp. 305–324], we have determined the leading term for the asymptotics of the eigenvalues λ of (P). In the present paper, we obtain, under more stringent assumptions, the corresponding remainder estimates. More precisely, let N ^±(λ) be the number of positive (respectively, negative) eigenvalues of (P) less than λ>0 (respectively, greater than λ<0); set r _± = max (±r, 0) and \(\Omega _ \pm = {\text{\{ }}x \in \Omega :r{\text{(}}x{\text{)}} \gtrless {\text{0\} }}\) . We show that $$N^ \pm {\text{(}}\lambda {\text{) = }}\mathop \smallint \limits_{\Omega _ \pm } {\text{(}}\lambda r{\text{(}}x{\text{))}}^{\frac{k}{{{\text{2}}m}}} {\text{ }}\mu \prime _\mathcal{A} {\text{(}}x{\text{) }}dx + 0{\text{(}}\left| \lambda \right|^{\frac{{k - 1}}{{{\text{2}}m}} + \delta } {\text{) as }}\lambda \to \pm \infty {\text{,}}$$ , where δ>0 and μ^′ _A (x) is the Browder-Gårding density associated with the principal part of A. How small δ can be chosen depends on the “regularity” of the leading coefficients of A, r _±, and of the boundary of Ω _±. These results seem to be new even for positive weights.     

The essential self-adjointness of Schrödinger-type operators

The paper discusses conditions under which the formally self-adjoint elliptic differential operator in R ^m given by 1 $$\tau {\text{ }}u = \sum\limits_{j,{\text{ }}k = 1}^m {[i\partial _j + b_j (x)]} {\text{ }}a_{jk} (x){\text{ }}[i\partial _k + b_k (x)]{\text{ }}u + q(x){\text{ }}u$$ has a unique self-adjoint extension. The novel feature is that the major conditions on the coefficients have to be imposed only in an increasing sequence of shell-like regions surrounding the origin. On the other hand it is shown that if these shells are broken so as to allow a tube extending to infinity in which the conditions on the coefficients are too weak, then, regardless of the coefficients elsewhere, there may not be a unique self-adjoint extension. The mathematical theorems are linked to the quantum-mechanical interpretation of essential self-adjointness (in the case that τ is the Schrödinger operator), that there is a unique self-adjoint extension if the particle cannot escape to infinity in a finite time.     

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      19. Generalized Flows Satisfying Spatial Boundary Conditions      B. Buffoni 《Journal of Mathematical Fluid Mechanics》2012,14(3):501-528 In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.      20. On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term      I. Peral  J. L. Vazquez 《Archive for Rational Mechanics and Analysis》1995,129(3):201-224 We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ 0=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.