# 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.

2,855
questions

**1**

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14 views

### Uniqueness for a transport-diffusion equation with low integrable drift

Consider the equation
$$
\frac{\partial f}{\partial t} + u \cdot \nabla f - \Delta f = 0
$$
in $(0,T) \times \mathbb R^N$, with initial condition
$$
f \vert_{t=0} = f_0
$$
for some given $f_0 \colon \...

**0**

votes

**0**answers

46 views

### Relation between the $L^2$ norm of the Poisson bracket of $f$ and $g$ and their $H^1$ norms

Let $f,g\in H^1(\Omega)$ where $\Omega$ is a sufficiently nice bounded domain in $\mathbb{R}^2$. If $\{\cdot,\cdot\}:H^1(\Omega)\times H^1(\Omega)\to L^2(\Omega)$ is the Poisson bracket, is there some ...

**6**

votes

**3**answers

300 views

### about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let f ...

**4**

votes

**1**answer

187 views

+50

### A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...

**2**

votes

**1**answer

96 views

### Gronwall estimate with a Fourier transform

Suppose I have the following equality
$$\hat{u}_\epsilon(t,k) = \alpha(t,k) + \int_0^t\int_{\mathbb{R}^n} e^{ik\cdot(x +\epsilon \phi(s,x))}u_\epsilon(s,x)dxds$$
Where $\alpha(t,k) \geq 0$ and $\alpha(...

**3**

votes

**0**answers

134 views

### Nonconstant smooth maps $f:\mathbb{R}^4-\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$

Does there exist a nonconstant smooth map $f:\mathbb{R}^4\setminus\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$ such that each $f(x):\mathbb{R}^4\to\mathbb{R}^5$ is an isometric linear map and $x\...

**1**

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**0**answers

66 views

### Does my functional satisfy the Palais Smale condition?

Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...

**0**

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**0**answers

67 views

### Linearization Navier-Stokes

I am considering the classical form of the stationary Navier-Stokes equation given by
$$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...

**1**

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**0**answers

63 views

### Behavior of the principal eigenfunction of fractional Laplacian

How does the first eigenfunction $\phi_{1}$ behave near the boundary of $\Omega$ where $$(-\Delta)^s\phi_{1}=\lambda_{1}\phi_{1},\text{ in } \Omega; \phi_{1} =0 \text{ in } \Omega^c$$ in a n-...

**0**

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**0**answers

14 views

### Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic

I first asked this question on math.stackexchange as I first thought it was somewhat innocent.
I would like if one can solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where
$...

**0**

votes

**1**answer

99 views

### Asymptotics for solution of transport equation and characteristics

Consider the transport equation $$u_t(t,x) + v(t,x) \cdot \nabla u(t,x) = 0.$$
Suppose that the solution of the characteristic equation
$$\dot X(t) = v(t,X(t)) $$
decays to zero as $t \to \infty$. ...

**-1**

votes

**0**answers

46 views

### transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$

Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$.
Applying the transformation $\Phi (\psi)=e^{...

**0**

votes

**0**answers

29 views

### Boundary regularity of rectifiable multiplicity 1 hypercurrents

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...

**2**

votes

**0**answers

73 views

### Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem

I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...

**2**

votes

**2**answers

113 views

### Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation
$$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$
with initial condition $u(0, x) = g(x).$
Suppose that $c(t, x)$ and $...

**2**

votes

**0**answers

30 views

### On maximum principle of spectral fractional Laplacian

Suppose $(-\Delta)^s u=g \geq 0$ in $\Omega$ and $u=0$ in $ \partial \Omega.$ Also suppose $u$ is $C^{2}$ non-negative and $(-\Delta)^s u=0$ in $\Omega \setminus B$ and $u\leq a$ on $\partial B $ ...

**2**

votes

**0**answers

51 views

### Itō formula for the solution of a SPDE in the distributional sense

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...

**1**

vote

**1**answer

101 views

### first order derivative of the parabolic equation

Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE
$$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...

**1**

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**0**answers

34 views

### Modelling fluid flows with mean curvature flow

A while ago I was wondering if the displacement of fluid described in this blog post
could be modelled with mean curvature flow or some other flow, but when I asked someone in Engineering they replied ...

**3**

votes

**0**answers

40 views

### Harnack inequality for $F(D^{2}u, Du,x)=f$

I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$, ...

**5**

votes

**1**answer

278 views

### When is a distribution having a finite support actually zero?

Let $D$ be a differential operator with smooth coefficients in $\mathbb{R}^n$. Suppose $E$ is a bounded open set in $\mathbb{R}^n$. Suppose $u$ is a function that is smooth up to the boundary of $E$. ...

**3**

votes

**0**answers

46 views

### Complex Monge Ampere equal with zero right hand side

Let $(X, \omega)$ be a compact Kaehler manifold, with $K_{X}$ numerically effective. Suppose that $[\zeta] = c_1(K_X)$ and $\int_{X}\zeta^n = 0$. I am interested in solving
\begin{equation}
(\zeta + i ...

**3**

votes

**1**answer

91 views

### Existence of a special function

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.
Is there any smooth function $...

**2**

votes

**0**answers

85 views

### Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...

**5**

votes

**1**answer

172 views

### Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...

**3**

votes

**1**answer

117 views

### Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...

**2**

votes

**0**answers

48 views

### Modified variational formulation of heat equation

The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to
$$
u_t = \Delta u,
$$
subject to certain boundary conditions and can alternatively be described, in variational form, as ...

**2**

votes

**0**answers

72 views

### Can Schauder's fixed point theorem apply to a metric space?

I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...

**5**

votes

**0**answers

98 views

### Sobolev extension from a discrete set of points

Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...

**1**

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**0**answers

71 views

### Change of one boundary condition in a problem

This is a problem in modeling in hydraulic fracturing field. It's quite long so hopefully someone can patiently read and help me.
The equation numbers are match those of the reference paper by ...

**5**

votes

**0**answers

83 views

### Relations between $\mathcal D$-modules and Exterior Differential Systems?

As a followup of this answer, I wonder relations between $\mathcal D$-modules and Exterior Differential Systems, both of which could be used to describe systems of PDEs. If I understand correctly, $\...

**30**

votes

**2**answers

1k views

### Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where ...

**15**

votes

**4**answers

4k views

### unique continuation principle

I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on ...

**6**

votes

**0**answers

321 views

### Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...

**3**

votes

**1**answer

78 views

### How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by
$$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...

**10**

votes

**3**answers

817 views

### Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...

**0**

votes

**0**answers

22 views

### Some scattering properties of cubic NLKG (how can we check the mixed-norm is bounded)?

Let $n=3$ and $u$ be the solution to Klein-Gordon equation
\begin{equation}
\begin{cases}\dot{u}-\Delta u +u=u^3 \\
u(0)=u_0, \partial_t u(0)=u_1,
\end{cases}
\end{equation}
where $(u_0,u_1) \in H^...

**3**

votes

**0**answers

106 views

### Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...

**2**

votes

**0**answers

66 views

### Reference for Rellich Kondrachov theorem on bounded domains and spaces with finite measure

I read at the top of the page $580$ of this paper that the imbedding $W^{1,2}(\Gamma,d\mu) \hookrightarrow L^2(\Gamma,d\mu)$ is compact for a bounded domain $\Gamma \subset \mathbb{R}^n$ and a measure ...

**0**

votes

**1**answer

143 views

### How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...

**1**

vote

**1**answer

71 views

### A completion equality in product of Sobolev spaces

Let $\Omega$ be a nice bounded domain of $\mathbb{R}^n$. For $s\ge 0$ we define
$$H_{0}^{s}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{H^{s}(\Omega)}.$$
My question: is the following equality true?
$$...

**11**

votes

**1**answer

1k views

### A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...

**1**

vote

**0**answers

46 views

### Proof of uniqueness of solution of the Poisson's equation for given boundary conditions [closed]

I asked this question last week on Mathematics Stack Exchange, but I didn't get any feedback. It is not a research level question, however, I post it here, perhaps somebody could help me:
I would ...

**1**

vote

**0**answers

43 views

### propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...

**2**

votes

**0**answers

44 views

### Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain
$$
\begin{cases}
\rho_t + q_x = 0 \\
q_t + (q^2/\rho + \rho)_x = - q \\
u|_{t=0} = u_0 \\
u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t)
\end{cases}
$$
in which ...

**2**

votes

**1**answer

143 views

### Numerical analysis of parabolic obstacle problem

I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$
\begin{align*}
\langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\...

**2**

votes

**1**answer

70 views

### Explicit solution of a free boundary problem for heat equation

Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...

**1**

vote

**0**answers

84 views

### Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$

Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...

**0**

votes

**0**answers

53 views

### Inf sup condition for discrete Stokes problem

I am considering the discrete inf-sup condition for the discrete Stokes problem
$ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$
This means ...

**2**

votes

**0**answers

41 views

### $C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

Consider a Fokker-Planck equation:
$$
\partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0,
$$
with initial condition ...