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.
1,733
questions with no upvoted or accepted answers
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95
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Partial regularity of harmonic maps into spheres
Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
3
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64
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Schrödinger equation: well-posedness with Hartree potential and Yukawa potential
Consider the Schrödinger equation of Hartree type (HT):
$$i\partial_tu +\Delta u + (V\ast |u|^2)u=0, u(x,0)=u_0$$
with $(x,t)\in \mathbb R^d \times \mathbb R.$
where $V$ is some potential.
(1) when $...
3
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0
answers
345
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On Solving a Fourth-Order Non-Linear PDE
I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity at the leading edge of a flat plate when fluid flows past it. The ...
3
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0
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346
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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 ...
3
votes
0
answers
336
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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 ...
3
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answers
102
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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 system
$$v_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$
whose $W_p^...
3
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0
answers
47
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How to prove that most of the systems of PDEs cannot be decoupled?
Let's consider the following system
\begin{align}
\frac{d}{dt}\begin{pmatrix}
u\\v
\end{pmatrix}=\begin{pmatrix}
f(x,t,u,v,u_x,v_x)\\g(x,t,u,v,u_x,v_x)
\end{pmatrix}.
\end{align}
If there locally ...
3
votes
0
answers
156
views
Perturbation theory compact operator
Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...
3
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answers
167
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Hardy-Littlewood in Sobolev Spaces
For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
3
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answers
55
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system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
3
votes
0
answers
191
views
Singular integral operators and PDEs
What is the relation between the notion of singular integral operators and partial differential equations?
I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
3
votes
0
answers
86
views
Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain
Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
3
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answers
93
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Green's function of time-dependent Stokes equation
It is well known that the Green's function of a standard parabolic equation in a bounded domain, say
\begin{align}
\partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0
&&\mbox{for}\,\,\,(x,t)\in \...
3
votes
0
answers
127
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Unique continuation from the boundary for inhomogeneous elliptic pde
Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
3
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answers
162
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Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
3
votes
0
answers
100
views
Ratio of solutions to two heat equations
Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
3
votes
0
answers
158
views
Fractional embedding inequality with $L^{\infty}$ norm
Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. For $q>2$, is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert ...
3
votes
0
answers
101
views
Inequality concerning BV norm
Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...
3
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216
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Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
3
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answers
108
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Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?
Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set.
Is it true that $(S,w)$ is metrizable?
Since $S$ is relatively weakly compact, it ...
3
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0
answers
181
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Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?
Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...
3
votes
0
answers
112
views
Parametrix of external product of elliptic operators
Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
3
votes
0
answers
82
views
Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$
Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...
3
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0
answers
79
views
Generalized viscosity sub(super)solution and it's convolution
Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant.
Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...
3
votes
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answers
84
views
Estimate a function given an estimate of its Laplacian
Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions:
\begin{eqnarray*}
\int_B |f_\lambda(x)|^2dx\leq 1,...
3
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276
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The uniqueness of fundamental solution on $\mathbb R^n$?
Let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\mathbb R^n$ which satisfies for all $y\in \mathbb R^n$:
$(i) (\partial_t -\Delta_x )p_t(x,y)=0, \text{ }t>0, x\in \...
3
votes
0
answers
536
views
Propagation of Singularities
I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the ...
3
votes
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answers
350
views
Proving that system is Hamiltonian
This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-...
3
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197
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Reference request on connection between PDE problems
I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
3
votes
0
answers
95
views
Tchebychev coordinates on hyperbolic space
Is there any known example of an open subset $U\subset\mathbb{H}^n$ with infinite volume admitting (global) Tchebychev coordinates for $n\geq3$?
3
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answers
353
views
Reference on semigroup theory and fractional heat equation
Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.
Where can I find a ...
3
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answers
237
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Maximum principle for parabolic PDE's
I am reading Cao's paper about his proof on Calabi-Yau theorem and I am having two little questions that I shall post separately.
Let $M$ be a complex compact manifold of dimension $n$ and consider ...
3
votes
0
answers
169
views
Origin of the terminology "trace operator" related to boundary-value problems for PDEs
Important results in the theory of PDEs regarding boundary-value problems are trace and extension theorems. Since the trace operator essentially acts by restriction to the boundary of the domain, I ...
3
votes
0
answers
114
views
Degenerate Beltrami equation and inverse
The Beltrami equation $f_{\bar{z}}=\mu(z)f_{z}$ is degenerate when $\left \| \mu \right \|_{\infty}=1$. For these equations, Lehto and David among others have given conditions for existence. The Lehto ...
3
votes
0
answers
120
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
3
votes
0
answers
150
views
Wolff's article: Note on counterexamples in strong unique continuation problems
I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, ...
3
votes
0
answers
124
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Exponentially weighted spaces: Effect on spectrum
My question is somewhat broad, but I do not know how to precisely state the issue. I am investigating stability of certain class of scalar PDE on $\mathbb{R}$.
Previous work in this topic has ...
3
votes
0
answers
145
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When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
3
votes
0
answers
192
views
Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space
Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
3
votes
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answers
140
views
Prove the positivity of the subelliptic operator heat kernel
Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a ...
3
votes
0
answers
278
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
3
votes
0
answers
96
views
Uniqueness of 1-dimensional heat-equation with dynamic boundary condition
My question regards uniqueness of the pair $(u(t, x), v(t))$ which
satisfy the following one dimensional time dependent heat equation
with a(n) (also time varying) Robin boundary condition at the ...
3
votes
0
answers
81
views
Books on turbulent compressible fluid (gas) in heated channel
It's been a while I got my hands dirty with simulation of hydrodynamics and it was mostly incompressible and laminar. Now, I need to model turbulent flow in channel with additional external heating ...
3
votes
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124
views
Wave equation that becomes elliptic on a bounded domain (sign-changing coefficient)
I'm looking for results on this kind of problems:
$$ \partial_{tt}^2 u - \partial_x(a(x) \partial_x) = f,$$
$$u(t=0) = u_0, \quad \partial_t u(t=0) = u_1,$$
where $a$ changes sign: $a(x)= -c^2 < 0$ ...
3
votes
0
answers
99
views
Eigenvalue estimate of a Neumann-like Laplacian in a Lipschitz domain
I have read quite some references on the smallest (for Dirichlet) and second smallest eigenvalues for the following Neumann eigenvalue problem:
$$
\begin{cases}
-\Delta u = \lambda u,\;\text{ in } \...
3
votes
0
answers
198
views
Analytic solution to two component, first order, linear PDE system
I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
3
votes
0
answers
202
views
Reference: Differential geometry on surfaces that are graphs of 2D-fluid-equations and Point Vortices
In "Pressure Field, Vorticity Field, and Coherent Structures in
Two-Dimensional Incompressible Turbulent Flows" Larchev$\hat{e}$que computes the Gaussian curvature of the surface of the streamfunction ...
3
votes
0
answers
108
views
Decay of frequencies of solution of the heat equation with a potential
Let $I_k = \{f \in L^2(R^n); supp(\hat{f}) \subset B(0,k) \}$.
Let $\Pi_k$ the orthogonal projection on $I_k$.
Let $a(t,x)$ a regular bounded potential.
Let $f$ the solution in $L^2$ to the Cauchy ...
3
votes
0
answers
1k
views
About Frobenius's theorem for differential forms
My question is about a particular case of Frobenius's theorem that states the complete integrability condition for a Pfaff system. Namely,
when dealing with a system reduced to a single 1-form, the ...
3
votes
0
answers
535
views
Time-dependent Sobolev spaces
Given the Sobolev space $H^1((a,b);H^2(\mathbb{R}))$ and a function $g$ in that space. Consider now another function $f \in C_c^{\infty}((a,b) \times \mathbb{R}).$ Then
for almost any $t \in (a,b)$ we ...