# Tagged Questions

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### 2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...
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### What to read for many-body problems in 3D Schrodinger equation

I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE ...
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### Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
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### Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
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### A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus: $\nabla^2 V = \frac{f(V)}{R^2}$ ...
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### Do circular pipes maximize flow rate?

Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...
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### Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation $$u_t-iu u_x-iu_{H,xx}=0$$ where $u_H(x,t)$ denotes the ...
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### Exact solutions to nonlinear Klein-Gordon equation

The nonlinear pde $$\partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0$$ has the exact solution $$\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i)$$ ...
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### A heat kernel for Schrödinger operator with low-order terms

In "SchrÃ¶dinger Operator: Heat Kernel and Its Applications", Feng computes the heat kernels associated to SchrÃ¶dinger operators with at most quadratic potentials. I am trying to see how these work ...
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### Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
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### Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
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### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
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### Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0$ for $\left | \sigma \right |\leqslant ... 1answer 280 views ### Linearization instability and singular points of algebraic varieties In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ... 3answers 508 views ### Space of sections of a fibre bundle with non-compact base space Let$\pi: E \rightarrow M$be a fiber bundle over the manifold M and denote by$\Gamma(E)$the space of smooth sections of$E$. For compact$M$it is well known (Hamilton 1982, Part II Corollary ... 0answers 114 views ### damped wave equation For$t>0$,$x$in a compact Riemannian manifold$(M,g)$, and$a\in C^\infty(M)$,$a\geq0$,$(\partial_t^2+a\partial_t-\Delta_g)u=0$is called the damped wave equation. My question is...why is the ... 1answer 123 views ### Distributional limits concerning the regularity of Maxwells equations This question is related to my previous question about the regularity of the Maxwell equations. Assume we are working on a space where there are only electric point charges,$(q_i)$, and a blob of ... 16answers 5k views ### Does Physics need non-analytic smooth functions? Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ... 1answer 491 views ### Solution of Helmholtz-Equation where Phase is restricted by additional PDE Hello! Let's say I have$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$with$f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ... 0answers 132 views ### Weyl quantization and convexity Let$C$be a convex subset of$\mathbb R^{2n}$and$\mathbf 1_C$be the characteristic function of$C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf ... 1answer 123 views ### A special Integral Kernel Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ?? f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy ... 0answers 205 views ### How can we formulate maximal time T in Hyperbolic Kahler Ricci ï¬‚ow In general, the exact maximal time T of a Riemannian Ricci flow may not be easy to find. However, fortunately, for KÃ¤hler-Ricci flows, the maximal time of existence T is explicitly determined by ... 2answers 418 views ### Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model Let us suppose that there exists a (initially smooth) solution of NSE that blows up in finite time. Then, in particular, the corresponding velocity field becomes unbounded as time progresses. Which ... 1answer 866 views ### Short time existence on nonlinear parabolic PDE I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ... 1answer 389 views ### Can one understand the Kelvin transform conceptually? Let U = \mathbf{R}^n - \{ 0 \}, n > 2 and consider for a function f \in C^2(U) the Kelvin transform$$f^\star(x) = r^{2-n} f\left(\frac{x}{r^2}\right),$$where r = \lvert x \rvert. One ... 1answer 102 views ### Is C_c(\mathbb{R}^2,\mathbb{R}^2) dense in the irrotational square integrable functions? Let L_D(\mathbb{R}^n)^n be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is$$L_D(\mathbb{R}^n)^n=\{Du\colon u\in ... 2answers 887 views ### What is soliton I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking.. In literature, i am finding following words:(Wikipedia+ others). ... 2answers 498 views ### Blow-up for the quasilinear heat equation$u_t= u \ u_{x x}$or the related$w_t= \left(w_x e^w\right)_x$What kind of approaches can be used to study the following quasilinear parabolic pde for a scalar function$u=u(x,t)$? $$u_t= u \ u_{x x}$$ The physical problem where this pde comes from dictates ... 1answer 170 views ### Curvatures of contours of solutions of 3d Poisson's equation Let$f(x,y,z)\$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ... 2answers 481 views ### Flow of evolutionary vector fields Consider a smooth vector bundle \pi: E\rightarrow M, the associated infinite jet bundle J^\infty(\pi), and evolutionary vector fields \partial_\varphi = ... 2answers 442 views ### Is there a spectral theory approach to non-explicit Plancherel-type theorems? Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on \mathbb{R}. There are ... 3answers 370 views ### Hard-sphere gases and the wave equation I'm trying to bridge a basic gap in my own education: Where can I find a written discussion concerning the derivation of the wave equation (for the propagation of sound, say), assuming nothing ... 1answer 338 views ### Regarding Discrete Eigenvalues For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete. But, ... 1answer 427 views ### Asymptotics of the TBA equation The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is ... 1answer 517 views ### How does electric potential relate to mean curvature? Consider a compact, convex domain \Omega \subset \mathbb{R}^3 with |\Omega|=1 with smooth boundary \partial \Omega. Now consider the electric potential generated by this uniform mass ... 1answer 315 views ### Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient? Hello, I am considering the following non-linear heat equation$$ \left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R_+\times R^d $$where ... 0answers 240 views ### Geometric description of Jacobi's theorem on complete integrals of HJ eqn. I am not sure if this question is adapted to this site, if it is not, then I will delete it. The Hamilton--Jacobi theory is about the connection between: the solutions of an Hamilton--Jacobi ... 1answer 448 views ### A name for PDE systems which are neither under- nor overdetermined? The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither ... 2answers 204 views ### Poisson equation in the plane Hello, as I'm not an analyst, I'm having difficulties with the following, certainly well-known problem: one is given the PDE \Delta u(x,y)=\sqrt{x^2+y^2} in the "region" x^2+y^2\leq1 with the ... 2answers 443 views ### Localization of Laplacian eigenfunction on the unit square? Let A be the unit square, \{u_k\} is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, C_V = \inf\limits_k ... 0answers 338 views ### problem with non linear pde I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...
hi there, i have the following pde: $$(\partial_x x^4 \partial_x - \partial_t^2)y(x,t)=0$$ and found the solution $$y=a+t^2-1/x^2$$, with a a constant. Is this solution unique? Does anyone know of any ...