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      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.

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      Mean field games approximate Nash equilibria

      I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf. I have a question about a step in theorem 3.8 on page 17. Let ...
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      Existence and uniqueness for semilinear problem

      Consider the following problem: $$-\Delta u + [(u)^+]^\alpha = 0,$$ where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
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      Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian

      What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
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      52 views

      Weak derivative under the integral sign

      Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...
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      Computing spectra without solving eigenvalue problems

      There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
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      Asymptotically periodic potentials

      Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?
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      38 views

      Smooth function whose average value on any ball $B(x,r)$ is a polynomial of $r$

      Let $f$ be a smooth function on $\mathbb R^N$, assume that for any $x \in \mathbb R^N$, $\frac{1}{Vol(B(x,r))} \int_{B(x,r)}f$ is a polynomial of $r$ (we denote the polynomial by $p_x(r)$), where $B(...
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      How to find the implicit solution of PDE's?

      I wanted to know if there were hints and educated guesses to find the implicit solutions of complicated PDE's. I'm currently dealing with functions of the form $f(r,t)$. I know that a lot of time a ...
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      64 views

      Harnack inequaliity for the fractional Laplacian

      I would like to know if there exist a Harnack type inequality for the non-local operator of the form $$(-\Delta)^s u= f \text{ in } B\subset \mathbb R^N$$ with $0<s<1$ and $N \geq 1.$ Here $B$ ...
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      PDE's : diffusion equation : polynomial diffusion coefficient

      I'd like to find analytical solutions of that kind of differential equations : $$\partial_t c = \partial_x (D(c)\partial_x c) $$ with $D(c)$ a polynomial. The trivial cas $D(c)=a$ with $a$ a ...
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      A good book for the p-fractional Laplacian

      Can anyone suggest a good book or a nice self contained article for the p-fractional Laplacian for beginners. I am interested in the existence, regularity of weak solution of the following equation $$...
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      219 views

      Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

      For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
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      Look for a suitable cut-function: from Pierre Grisvard “Elliptic Problems in Nonsmooth Domains”: (Theorem 1.4.2.4)

      From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains": Theorem[Theorem 1.4.2.1] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\...
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      35 views

      Enhanced dissipation for Kolmogorov flow

      My problem is $$\frac{\partial u}{\partial t}+\ sin(y)\frac{\partial u}{\partial x}=\nu(\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2})$$ with periodic boundary conditions and ...
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      123 views

      Lax Milgram for non coercive problem?

      I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have $$a(u,v)=\int_\Omega u(t)...

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