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[PDE]Exercise 0001 Let solve the following initial-value problem

Show that has the representation . Here, is defined through the Fourier transform by . Solution

[PDE]Exercise 0002 Suppose that the function is radial, that is, for any with , we have . Show that the Fourier transform is also radial. Solution

[PDE]Exercise 0003 Let be Banach spaces, be a linear map. is said to be bounded, if , such that . Show that is bounded iff (if and only if) for any bounded subset , is a bounded subset of . Solution

[PDE]Exercise 0004 (1) Give the definition of the semi-norm and , where , . (2) Show that and are equivalent. Solution

[PDE]Exercise 0005 Let be in , and is radial (for definition, see Problem 2). Show that

Solution

[PDE]Exercise 0006 Let be smooth vector field. Show that . Solution

[PDE]Exercise 0007 Show the Bony decomposition

where

Solution

[PDE]Exercise 0008 For any positive , we have

Solution

[PDE]Exercise 0009 If is a smooth homogeneous function of degree , show that

where

is the Hardy-Littlewood maximal function. Solution

[PDE]Exercise 0010 (1) Narrate the resonance theorem. (2) Let be a Banach space, and denote by be all the maps

such that for any functional , the function is continuous. Utilize (1) to show . Solution

[PDE]Exercise 0011 Consider the three-dimensional Navier-Stokes equations

Let . Then by the Leray's famous work, there exists at least one weak solution of (1). Suppose that and

Show that is a strong solution, i.e., . This is the classical Ladyzhenskaya-Prodi-Serrin condition. Solution

[PDE]Exercise 0012 (1)、 As is well-known, the volumes of -dimensional unit ball are respectively. Then what is the volume of -dimensional unit ball

Provide the calculations. (2)、 Denote by the area of the -dimensional unit sphere

What is the relationship between and ? Solution

[PDE]Exercise 0013 (1)、 Let . Provide the definition of the convexity of . (2)、 As is well-known, a convex function has a supporting hyperplane at each point ; that is,

Now, let , , . Describe the set of all such satisfying (*). (3)、 Prove the Jensen inequality. Assume is convex and is open, bounded. Let be summable. Then

Solution

[PDE]Exercise 0014 (1)、 State the Young inequality. Let . Then

(2)、 Let satisfy

Use the Young inequality to show that

Solution

[PDE]Exercise 0015 (1)、 State the H"older inequality. Assume that

Then if , we have

(2)、 Assume and . Use the H"older inequality to show the following Minkowski inequality

Solution

[PDE]Exercise 0016 Let be a nonnegative, absolutely continuous function on , which satisfies for the differential inequality

where and are nonnegative, summable functions on . Then

for all . Solution

[PDE]Exercise 0017 Let be a nonnegative, absolutely continuous function on , which satisfies for the integral inequality

where and are nonnegative, summable functions on . Show that

Solution

[PDE]Exercise 0018 (1)、 State the Gauss-Green Theorem. Suppose . Then

(2)、 Use (1) to prove the following integration by parts formula. Let . Then

Solution

[PDE]Exercise 0019 (1)、 State the divergence theorem.

for each vector field . (2)、 Use the divergence theorem to show the Green's formula. Let . Then

Solution

[PDE]Exercise 0020 (1)、 Let be the standard mollifier. Then for locally integrable function , its mollification is

(2)、 If , then . Solution

[PDE]Exercise 0021 (1)、 Let be the standard mollifier. Then for locally integrable function , its mollification is

(2)、 If and , then in . Solution

[PDE]Exercise 0022 A mapping is called a norm if (1)、 for all , (2)、 for all , (3)、 iff (if and only if) . Solution

[PDE]Exercise 0023 A mapping is called an inner product if (1)、 for all , (2)、 the mapping is for each , (3)、 for all , (4)、 iff . Solution

[PDE]Exercise 0024 (1)、 Let be a Banach space. We say a sequence converges weakly to , written

if

for each bounded linear function . (2)、 Show that

Solution

[PDE]Exercise 0025 (1)、 A set is weakly closed if

(2)、 Assume is weakly closed. Show that is closed. Solution

[PDE]Exercise 0026 Let denote an open subset of , with a smooth boundary . Suppose , . Prove is a Banach space. Solution

[PDE]Exercise 0027 Let denote an open subset of , with a smooth boundary . Assume . Prove the interpolation inequality

Solution

[PDE]Exercise 0028 Let be open sets, with . Show there exists a smooth function such that on , near . Solution

[PDE]Exercise 0029 Assume is bounded and . Show there exist functions such that

The functions form a partition of unity. Solution

[PDE]Exercise 0030 Integrate by parts to prove

for and . Solution

[PDE]Exercise 0031 Prove