# 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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### Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}$$ over all ...
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### Linearization of a PDE

I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us ...
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### Question on Sobolev spaces in domains with boundary

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ ...
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### Exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han, about application of strong maximum principle

I have a question about exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han. Let assume $‎‎\Omega ‎\subset ‎‎\mathbb{R}^n‎$ is a bounded domain and $f$ and $u_0$ ...
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### Initial-boundary value problem for the damped wave equation with nonlinear source (in bounded domains)

It is well known that the initial-value problem for the wave equation on $\mathbb R^N$ can be studied by means of Fourier transform. What reference presents well-posedness results and qualitative ...
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### Is the minmod limiter energy stable?

It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
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### Long time existence for heat flow in Corlette-Donaldson Theorem

I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14) https://arxiv.org/pdf/1402.4203.pdf For completeness, the statement is as follows. ...
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### Wellposedness results for the cubic Schödinger equation

Motivated by the question Relationship between the vortex filament equation and the cubic Schrödinger equation, I'd like to ask the following: Where can I find a reference on wellposedness ...
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### Relationship between the vortex filament equation and the transport equation

Let us consider the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$. How is the Cauchy problem for the ...
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### Different versions of strong maximum principle?

Let assume $$Lu=Au+a(x)u=f(x)$$ Where $A$ is elliptic operator. In Chapter 8 of Smoller book the strong maximum principle is as follow: On the other hand, the strong maximum principle of Han-Lin ...
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### Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation? In particular, I'm interested in the following topics: physical motivation; notion of solutions and ...
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### Interior smooth regularity

I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth ...
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### A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
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### Compute $\partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator. How can I compute the ...
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### Discrete Sobolev embedding

It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$ Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
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### Linear system formulation for a PDE with Neumann boundary condition

PDE with Dirichlet boundary condition can be written as a linear system: $Au(x)=f(x); \ \forall x \in \Omega$, s.t. $u(x)=g(x); \ \forall x \in \Gamma$. This can be solved for instance using the ...
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### Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
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### Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider the following boundary value problem for the heat equation: ...
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### Embeddings of Sobolev spaces on the unit disk

Is the space $W^{2,p}(\mathbf{U})$, compactly embedded to $W^{1,\infty}(\mathbf{U})$, where $\mathbf{U}$ is the unit disk.
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### Definition of $\epsilon$-upper envelope of a function

Let $u$ be a continuous function in an open set $\Omega\subset \mathbb{R}^{n}$, and let $H$ an open set such that $\bar{H}\subset \Omega$. We define , for $\epsilon >0$, the upper $\epsilon$-...
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Apologies if this is a simple question, but I've left PDE as a field and a friend recently asked me the following question regarding solutions $u$ and $v$ of a system of PDE. Consider \nabla \cdot (... 0answers 176 views ### Existence and uniqueness for reaction-diffusion equations I am interested in the following PDE on a d-dimensional torus \mathbb{T}^d \begin{align*} &\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\ & u(0)=u_0\in L_2 \end{align*} where the ... 1answer 125 views ### Spectrum of the Laplacian on the quotient of 3-sphere Given a finite subgroup \Gamma of O(4) acting freely on S^3, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric 2-tensor on S^3/\Gamma equipped with ... 1answer 117 views ### What Morrey and Campanato space characterize Morrey and Campanato space is some subspace of L^p. We know that for a bounded domain \Omega, L^p space characterize how the function blow up at some point. I want to know what Morrey and ... 2answers 222 views ### Bounded weak derivative Let f \in L^{\infty} be a function such that f and the weak derivatives D^{\alpha}f\in L^{\infty} exist for all \vert \alpha\vert\ge 2. Does this imply that also D^{\alpha}f with \vert \... 1answer 168 views ### An alternative representation of the principal symbol of the Laplace operator Assume that (M,g) is a n dimensional Riemannian manifold. We denote by \Delta, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? First ... 0answers 241 views ### Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form? I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ... 2answers 223 views ### Derivations of \chi^{\infty}(M) which are elliptic operator What is an example of a manifold M with \dim(M)>1 whose Lie algebra \chi^{\infty}(M) of smooth vector fields admit an elliptic operator D:\chi^{\infty}(M)\to \chi^{\infty}(M) such ... 0answers 51 views ### A variant to the Stokes system and Navier-Stokes equation The linearization of the Navier-Stokes equation (in a smooth bounded domain in dimensions 2 or 3) is the following non-stationary Stokes systemv_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$whose W_p^... 0answers 45 views ### Tartar wave cone — oscillations and ellipticity In a work of De Philippis and Rindler, I found the following passage. Questions. How did they show that \Lambda_{\mathscr A} contains all the amplitudes along which the system is not elliptic? ... 0answers 38 views ### Strichartz estimate on wave equation For standard wave equation$$\Box u=F, u(0,\cdot)=f, u_t(0,\cdot)=g$$we have Strichartz estimate by Ginibre,Velo and Keel,Tao. Now I am considering a problem$$\Box u+e(t)u=F, u(0,\cdot)=f, u_t(0,\...
Consider the transport equations $$(1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$(2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...
Consider the conservation law $$(\ast) \begin{cases} \partial_t u + \partial_x f(u) = 0, & (t,x) \in (0,T)\times \mathbb R \\ u(0,x) = u_0(x), & x \in \mathbb R \end{cases}$$ Under what ...