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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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### 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 ...
1answer
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### 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 $B$ ...
1answer
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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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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 $$... 1answer 219 views ### Is this a “contradiction” on stochastic Burgers' equation? How to understand it? For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ... 0answers 58 views ### Look for a suitable cut-function: from Pierre Grisvard “Elliptic Problems in Nonsmooth Domains”: (Theorem 1.4.2.4) From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains": Theorem[Theorem 1.4.2.1] Let \Omega be an open subset of \mathbb{R}^d with a continuous boundary, then C_c^\infty(\overline{\... 0answers 35 views ### Enhanced dissipation for Kolmogorov flow My problem is$$\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})$$with periodic boundary conditions and ... 0answers 123 views ### Lax Milgram for non coercive problem? I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let \Omega\subset\mathbb{R} be an open set. For u,v\in H^1_0(\Omega). I have$$a(u,v)=\int_\Omega u(t)...

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