# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2,855 questions
Filter by
Sorted by
Tagged with
223 views

### Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases}$$ where $\chi$ denotes the ...
124 views

### Positive part of Cauchy sequence of sobolev functions is again Cauchy

Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
44 views

### Hopf type lemma for fractional Laplacian

Let $0<s<1$ and $u\in C^s(\mathbb R^N).$ Does the Hopf type of maximum principle hold for s-super-harmonic function $(-\Delta)^su\geq 0$ in a smooth bounded domain $\Omega\subset \mathbb R^N.$
2k views

138 views

167 views

89 views

### Justification for uniqueness of solutions to dispersive PDE

For the sake of concreteness, we consider the linear Schrödinger equation $$\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).$$ The solution is typically obtained by taking the Fourier transform ...
50 views

### Extension of a compactly supported pseudo-differential operator

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
90 views

49 views

In J. L. - Lions book Quelques méthodes de résolution des problèmes aux limites non linéaires, the author proves the following lemma: Lemme 6.1: Let w be a function satisfying $w \in L^\infty(0,... 0answers 89 views ### Is there any characterization of polynomials in terms of asymptotic properties of Taylor coefficients? [closed] My formal question is Let$f(z):=\sum_{n=0}^{\infty} c_n z^n$be a formal power series. Is there any characterization of polynomials in terms of the asymptotic properties the sequence$(c_n)$? For ... 0answers 76 views ### Biharmonic heat flow on compact manifolds Consider$\partial _t u (t,x) = -\partial _x ^4 u$on a compact manifold, or even a special specific one like the torus. Are there any estimates on the Green function (bihamornic heat kernel), for ... 0answers 46 views ### Solution of$\vec{p}-$Laplace equation Let$\Omega \subset {\mathbb{R}^n}$is bounded domain with smooth boundary. We consider the bvp $$- \sum\limits_{I = 1}^n {{\partial _{{x_i}}}\left( {{{\left| {{\partial _{{x_i}}}u} \right|}^{{p_i} ... 1answer 283 views ### Lack of exponential L^2_{t,x} decay for a heat equation with growing coefficients Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below). Given T>0 and n \in \bf Z, consider the following ... 0answers 44 views ### On L^\infty norm of solutions to time dependent differential equations I am new to the theory of differential equations and weak solutions. I am looking for references regarding the analysis of the L^\infty norm of weak solutions to linear second order time ... 0answers 71 views ### divergence equation with prescribed normal trace Let \Omega \subset \mathbb{R}^n be a smooth domain and \nu be the outer unit normal to \partial \Omega. Given \phi \in L^{\infty}(\partial \Omega) such that \int_{\partial \Omega} \phi d\... 1answer 76 views ### Fractional super-harmonic functions Is this statement true. A bounded half-superharmonic function in \mathbb R is a constant. That is (-\Delta)^{1/2} u\geq 0 implies u\equiv 0. 1answer 71 views ### Conservated quantity and hyperbolic equation Given the hyperbolic Vlasov equation$$ \frac{\partial f }{\partial t} +v\nabla_x f + F(t,x)\nabla_vf =0$$where$f=f(t,x,v)$and$(t,x,v)\in \mathbb{ R}\times\mathbb{R}^{n}\times \mathbb{R}^{n} $. ... 0answers 42 views ### what classical PDEs have analytical expressions for soliton-like shape solutions but motionless? what classical PDEs have analytical expressions for soliton-like shape solutions but motionless? for example, KdV has analytical expressions of the kind (sech^2(x-vt)), but all of them are ... 1answer 74 views ### Smooth approximation of a subharmonic function in the barrier sense Let$f$be a continuous function on$\mathbb R^n$such that$\Delta f \ge 0$at a point$p$in the barrier sense. More precisely, for any$\epsilon>0$, there exists a smooth function$f_{\epsilon}$... 1answer 302 views ### Laplace spectrum of the$2$-Sphere [closed] The$2$-sphere$S^2$endowed with usual round metric, we have a Laplacian operator$\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$acting on functions. The eigenvalues of this ... 0answers 66 views ### About p-laplacian and variations Let$\Omega \subset \mathbb{R^{n}}$be a domain (open and connected set), for$p\geq 2$, the$p$-laplacian is defined by:$\Delta_p u= \operatorname{div} (|\nabla u|^{p-2} \nabla u)$, in non-... 1answer 262 views ### Sobolev embedding in the space of continuous functions [duplicate] Let$I = \mathbb{R}$and let$W^{1,2}(I,\mathbb{R})$be the Sobolev space of function from$I$to$\mathbb{R}$(one time weakly differentiable and contained in$L^{2}$) and$C^{0}(I,\mathbb{R})$be ... 0answers 47 views ### discrete Fourier transform for composition of differential operators on a grid This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1)$\hspace{.5in}u_t + \...
I've just read a review paper about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds. Briefly, if $\phi_\lambda$, $\lambda$ are an eigenpair for the Laplace-...